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\begin{document}

\author{P\'eter Major}
\title{MULTIPLE WIENER--IT\^O INTEGRALS \\
 with applications to limit theorems}
\subtitle{-- Lecture Note -- 849}
%\subtitle{-- Monograph --}
\maketitle

\frontmatter

\tableofcontents

\preface

One of the most important problems in probability theory is the
investigation of the limit distribution of partial sums of
appropriately normalized random variables. The case where the random
variables are independent is fairly well understood. Many results are
known also in the case where independence is replaced by an appropriate
mixing condition or some other ``almost independence'' property. Much
less is known about the limit behaviour of partial sums of really
dependent random variables. On the other hand, this case is becoming
more and more important, not only in probability theory, but also in
some applications in statistical physics.

The problem about the asymptotic behaviour of partial sums of dependent
random variables leads to the investigation of some very complicated
transformations of probability measures. The classical methods of
probability theory do not seem to work for this problem. On the other
hand, although we are still very far from a satisfactory solution
of this problem, we can already present some nontrivial results.

The so-called multiple Wiener--It\^o integrals have proved to be a
very useful tool in the investigation of this problem. The proofs of
almost all rigorous results in this field are closely related to this
technique. The notion of multiple Wiener--It\^o integrals was worked
out for the investigation of non-linear functionals over Gaussian fields.
It is closely related to the so-called Wick polynomials which can be
considered as the multi-dimensional generalization of Hermite polynomials.
The notion of Wick polynomials and multiple Wiener--It\^o integrals
were worked out at the same time and independently of each other.
Actually,  we discuss a modified version of the multiple Wiener--It\^o
integrals in greatest detail. The technical changes needed in the
definition of these modified integrals are not essential. On the other
hand, these modified integrals are more appropriate for certain
investigations, since they enable us to describe the action of shift
transformations and to apply some sort of random Fourier analysis.
There is also some connection between multiple Wiener--It\^o integrals
and the classical stochastic It\^o integrals. The main difference
between them is that in the first case deterministic functions are
integrated, and in the second case so-called non-anticipating 
functionals. The consequence of this difference is that no technical  
difficulty arises when we want to define multiple Wiener--It\^o 
integrals  in the multi-dimensional time case. On the other hand, 
a large class of nonlinear functionals over Gaussian fields can be 
represented by means of multiple Wiener--It\^o integrals.

In this work we are interested in limit problems for sums of 
dependent random variables. It is useful to consider this problem 
together with its continuous time version. The natural formulation 
of the continuous time version of this problem can be given by 
means of  generalized random fields. Consequently we also have 
to discuss some questions about them.

I have not tried to formulate all the results in the most 
general form. My main goal was to work out the most important 
techniques needed in the investigation of such problems. This 
is the reason why the greatest part of this work deals with 
multiple Wiener--It\^o integrals. I have tried to give a 
self-contained exposition of this subject and also to 
explain the motivation behind the results.

I had the opportunity to participate in the Dobrushin--Sinai 
seminar in Moscow. What I learned there was very useful also 
for the preparation of this Lecture Note. Therefore I would 
like to thank the members of this seminar for what I could 
learn from them, especially P.~M.~Bleher, R.~L.~Dobrushin 
and Ya.~G.~Sinai.

\medskip\noindent
{\it Preface to the Second Edition.}

\medskip\noindent
This text is a slightly modified version of my Lecture Note 
{\it Multiple Wiener--It\^o integrals with applications to 
limit theorems}\/ published in the {\it Lecture Notes in 
Mathematics}\/ series (number~849) of the Springer Verlag 
in~1981. I decided to write a revised version of this
Lecture Note after a special course I held about its subject 
in the first semester of the academic year~2011--2012 at the 
University of~Szeged. Preparing for this course I observed 
how difficult the reading of formulas in this Lecture Note 
was. These difficulties arose because this Lecture Note was 
written at the time when the \TeX{} program still did not 
exist, and the highest technical level of typing was 
writing on an IBM machine that enabled one to type beside 
the usual text also mathematical formulas. But the texts 
written in such a way are very hard to read. To make my 
text more readable I decided to retype it by means 
of the \TeX{} program. But it turned out that a real 
improvement of the text demands much more than producing
nice, readable formulas. To make a really better version 
of this work I also had to explain better the results and 
definitions together with the ideas and motivation behind 
them. Besides, I had to make not only more readable 
formulas, but also more readable explanations. The reader 
must see at each point of the discussion what is just 
going on, and why. In the new version of this work I 
tried to satisfy these demands. Naturally, I also
corrected the errors I found. At some points I had to
insert a rather long explanation in the proof, because
I met such a statement which seemed to be trivial at the 
first sight, but its justification demanded a detailed 
discussion. I hope that these insertions did not make 
the work less transparent.

There appeared many new results about the subject of this
Lecture Note since its first appearance. The question 
arose naturally whether I should insert them to the 
new edition of this work. Finally I decided to make 
no essential changes in the text, to restrict myself 
to the correction of the errors I found, and to give 
a more detailed explanation of the proofs where I felt 
that it is useful. In making such a decision I was 
influenced by a  Russian proverb which says: `Luchshe 
vrag khoroshego'. I tried to follow the advice of this 
proverb. (I do not know of an English counterpart of 
this proverb, but it has a French version: `Le mieux 
est l'ennemi du bien'.)

I made one exception. I decided to explain those basic 
notions and results in the theory of generalized functions 
which were applied in the older version of this work in an 
implicit way. In particular, I tried to explain with
their help how one gets those results about the spectral 
representation of the covariance function of stationary 
random fields that I have presented under the names 
{\it Bochner's theorem}\/ and {\it Bochner--Schwartz 
theorem.} This extension of the text is contained in 
the attachments to Chapters~1 and~3. In the first 
version I only referred to a work where these notions 
and results can be found. But now I found such an 
approach not satisfactory, because these notions and 
results play an important role in some arguments of 
this work. Hence I felt that to make a self-contained 
presentation of the subject I have to explain them in 
more detail.

The first edition of this Lecture Note appeared long time
ago, but the main question discussed in it, the  description
of the limit behaviour of appropriately normalized partial 
sums of strongly dependent random variables remained an
open problem. Also the method applied in this work 
remained an important tool in the study of such  
problems. Hence a self-contained explanation of the 
theory which provides a good foundation for this method
is useful. By my hopes this Lecture Note contains such 
an explanation, and therefore it did not become out of
date. This was the main argument for myself to write
a new version of this work where I tried to present a 
better and more accessible discussion. 

I would like to write some words about the last chapter of 
this work, where some results are discussed that seemed 
to be important at the time of writing the first version.
I would mention two of them which later turned out to be 
really important. The first one is the Nelson--Gross 
inequality which later played an important role in the 
theory of the so-called hypercontractive and logarithmic 
Soboliev inequalities. The second one is a method for 
construction of non-trivial self-similar fields worked 
out in a paper of Kesten and Spitzer. Several important 
limit theorems are based on the ideas of this paper. 
It is worth mentioning that it was Roland L'vovich 
Dobrushin who called my attention to these results, 
and he emphasized their importance. So I would like 
to finish this preface with a personal remark about him.

This work is the result of some joint research with 
Roland L'vovich Dobrushin. Although the book was 
written by me alone, Dobrushin's influence is very 
strong in it. I have learned very much from him. It is 
rather difficult to explain what one could learn from 
him, because it was much more than just some results 
or mathematical arguments. There was something 
beyond it, some world view which is hard to explain. 
If I could give back something from what I had 
learned from him in this Lecture Note, then this 
would justify the work on it by itself.

\bigskip\noindent
Budapest, 15 August 2013

\medskip
\hskip8truecm P\'eter Major

\mainmatter

\chapter{On a Limit Problem}

We begin with the formulation of a problem which is important 
both for probability theory and statistical physics. The multiple 
Wiener--It\^o integral proved to be a very useful tool at the 
investigation of this problem.

Let us consider a set of random variables $\xi_n$,
$n\in \textrm{\BBB Z}_\nu$, where $\textrm{\BBB Z}_\nu$ 
denotes the $\nu$-dimensional integer lattice, and let us 
study their properties. Such a set of random variables 
will be called a ($\nu$-dimensional) discrete random field. 
\index{discrete random field}
We shall be mainly interested in so-called stationary random 
fields. Let us recall their definition.

\medskip\noindent
{\bf Definition of Discrete (Strictly) Stationary Random Fields.} 
\index{stationary (discrete) random fields} 
{\it A set of random variables $\xi_n$, $n\in\textrm{\BBB Z}_\nu$, 
is called a (strictly) stationary discrete random field if
$$(
\xi_{n_1},\dots,\xi_{n_k})\stackrel{\Delta}{=}
(\xi_{n_1+m},\dots,\xi_{n_k+m})
$$
for all $k=1,2,\dots$ and \ $n_1,\dots,n_k,\;m\in\textrm{\BBB Z}_\nu$,
where $\stackrel{\Delta}{=}$ denotes equality in distribution.}

\medskip
Let us also recall that a discrete random field $\xi_n$,
$n\in\textrm{\BBB Z}_\nu$, is called Gaussian if for every finite
subset $\{n_1,\dots,n_k\}\subset\textrm{\BBB Z}_\nu$ the random
vector $(\xi_{n_1},\dots,\xi_{n_k})$ is normally distributed.

Given a discrete random field $\xi_n$, $n\in\textrm{\BBB Z}_\nu$, we
define for all $N=1,2,\dots$ the new random fields
\begin{equation}
Z_n^N=A_N^{-1}\sum_{j\in B_n^N}\xi_j, \qquad N=1,2,\dots, \quad
n\in\textrm{\BBB Z}_\nu,
\label{(1.1)}
\end{equation}
where
$$
B_n^N=\{j\colon\; j\in \textrm{\BBB Z}_\nu,\quad n^{(i)}N\le
j^{(i)}<(n^{(i)}+1)N,\;i=1,2,\dots, \nu\},
$$
and $A_N$, $A_N>0$, is an appropriate norming constant. The superscript
$i$ denotes the $i$-th coordinate of a vector in this formula. We are
interested in the question when the finite dimensional distributions 
of the random fields $Z_n^N$ defined in~(\ref{(1.1)}) have a limit as 
$N\to\infty$.
In particular, we would like to describe those random fields $Z_n^*$,
$n\in\textrm{\BBB Z}_\nu$, which appear as the limit of such random
fields~$Z_n^N$. This problem led to the introduction of the following
notion.

\medskip\noindent
{\bf Definition of Self-similar (Discrete) Random Fields.} 
\index{self-similar (discrete) random field} 
\index{self-similarity parameter} 
{\it A (discrete) random field 
$\xi_n$, $n\in\textrm{\BBB Z}_\nu$, is called self-similar 
with self-similarity parameter~$\alpha$ if the random
fields $Z^N_n$ defined in~(\ref{(1.1)}) with their help 
and the choice $A_N=N^\alpha$ satisfy the relation
\begin{equation}
(\xi_{n_1},\dots,\xi_{n_k})\stackrel{\Delta}{=}
(Z^N_{n_1},\dots,Z_{n_k}^N) \label{(1.2)}
\end{equation}
for all $N=1,2,\dots$ and $n_1,\dots,n_k\in\textrm{\BBB Z}_\nu$.}

\medskip
We are interested in the choice $A_N=N^\alpha$ with some 
$\alpha>0$ in the definition of the random variables~$Z^N_n$ 
in~(\ref{(1.2)}), because under slight restrictions, 
relation~(\ref{(1.2)}) can be satisfied only with
such norming constants $A_N$. A central problem both in statistical
physics and in probability theory is the description of self-similar
fields. We are interested  in self-similar fields whose random
variables have a finite second moment. This excludes the fields
consisting of i.i.d. random variables with a non--Gaussian stable law.

The Gaussian self-similar random fields and their Gaussian range 
of attraction are fairly well known. Much less is known about 
the non-Gaussian case. The problem is hard, because the 
transformations of measures over $R^{\textrm{\BBB Z}_\nu}$ 
induced by formula~(\ref{(1.1)}) have a very complicated 
structure. To get interesting results in some cases we shall 
define the so-called subordinated fields below. (More precisely, 
we define the fields subordinated to a stationary Gaussian field.) 
In case of subordinated fields the Wiener--It\^o integral is a 
very useful tool for investigating the transformation defined 
in~(\ref{(1.1)}). In particular, it enables us to construct
non--Gaussian self-similar fields and to prove non-trivial limit
theorems. All known results are closely related to this technique.

Let $X_n$, $n\in\textrm{\BBB Z}_\nu$, be a stationary Gaussian 
field. We define the shift transformations 
$T_m$, $m\in\textrm{\BBB Z}_\nu$, over this field by the formula 
$T_mX_n=X_{n+m}$ for all $n,\,m\in\textrm{\BBB Z}_\nu$. Let 
${\cal H}$ denote the {\it real} Hilbert space consisting of 
the square integrable random variables measurable with respect 
to the $\sigma$-algebra
${\cal B}={\cal B}(X_n,\;n\in\textrm{\BBB Z}_\nu)$. The scalar 
product in ${\cal H}$ is defined as $(\xi,\eta)=E\xi\eta$, 
$\xi,\,\eta\in{\cal H}$. The shift transformations $T_m$, 
$m\in\textrm{\BBB Z}_\nu$, can be extended to a group of 
unitary shift transformations over ${\cal H}$ in a natural way. 
Namely, if $\xi=f(X_{n_1},\dots,X_{n_k})$ then we define 
$T_m\xi=f(X_{n_1+m},\dots,X_{n_k+m})$. It can be seen that
$\|\xi\|=\|T_m\xi\|$, and the above considered random variables 
$\xi$ are dense in ${\cal H}$. 
\index{shift transformation (discrete random field)}
(A more detailed discussion about the definition of shift 
operators and their properties will be given in Chapter~2 in
a {\it Remark}\/ after the formulation of Theorem~2C. Here 
we shall define the shift $T_m\xi$, $m\in\textrm{\BBB Z}_\nu$, 
of all random variables $\xi$ which are measurable with respect 
to the $\sigma$-algebra ${\cal B}(X_n,\,n\in{\BBB Z}_\nu)$, i.e. 
$\xi$ does not have to be square integrable.) Hence $T_m$ 
can be extended to the whole space ${\cal H}$ by $L_2$ 
continuity. It can be proved that the norm preserving 
transformations $T_m$, $m\in\textrm{\BBB Z}_\nu$, constitute a 
unitary group in ${\cal H}$, i.e. $T_{n+m}=T_nT_m$ for all 
$n,\,m\in\textrm{\BBB Z}_\nu$, and $T_0=\textrm{Id}$. Now we
introduce the following

\medskip\noindent
{\bf Definition of Subordinated Random Fields.} 
\index{random field subordinated to a stationary Gaussian 
random field (discrete case)} 
{\it Given a stationary Gaussian field 
$X_n$, $n\in\textrm{\BBB Z}_\nu$, we define the Hilbert spaces
${\cal H}$ and the shift transformations $T_m$, 
$m\in\textrm{\BBB Z}_\nu$, over ${\cal H}$ as before. A 
discrete stationary field $\xi_n$ is called a random field 
subordinated to $X_n$ if $\xi_n\in{\cal H}$, and
$T_n\xi_m=\xi_{n+m}$ for all $n,\,m\in\textrm{\BBB Z}_\nu$.}

\medskip
We remark that $\xi_0$ determines the subordinated fields 
$\xi_n$ completely, since $\xi_n=T_n\xi_0$. Later we give a 
more adequate description of subordinated fields by means of 
Wiener--It\^o integrals. Before working out the details we 
formulate the continuous time version of the above notions 
and problems. In the continuous time case it is more natural 
to consider generalized random fields. To explain the idea 
behind such an approach we shortly explain a different but 
equivalent description of discrete random fields. We present 
them as an appropriate set of random variables indexed by 
the elements of a linear space. This shows some similarity 
with the generalized random fields to be defined later.

Let $\varphi_n(x)$, $n\in\textrm{\BBB Z}_\nu$, 
$n=(n_1,\dots,n_\nu)$, denote the indicator function of the 
cube
$[n_1-\frac12,n_1+\frac12)\times\cdots\times[n_\nu-\frac12,n_\nu+\frac12)$,
with center $n=(n_1,\dots,n_\nu)$ and with edges of length~1,
i.e.\ let $\varphi_n(x)=1$, $x=(x_1,\dots,x_\nu)\in R^\nu$, if
$n_j-\frac12\le x_j<n_j+\frac12$ for all $1\le j\le \nu$, and let
$\varphi_n(x)=0$ otherwise. Define the linear space $\Phi$ of
functions on $R^\nu$ consisting of all finite linear combinations of
the form $\sum c_j\varphi_{n_j}(x)$, $n_j\in\textrm{\BBB Z}_\nu$, 
with the above defined functions $\varphi_n(x)$ and real
coefficients~$c_j$. Given a discrete random field $\xi_n$,
$n\in\textrm{\BBB Z}_\nu$, define the random variables 
$\xi(\varphi)$ for all $\varphi\in\Phi$ by the formula
$\xi(\varphi)=\sum c_j \xi_{n_j}$ if
$\varphi(x)=\sum c_j\varphi_{n_j}(x)$. In particular,
$\xi(\varphi_n)=\xi_n$ for all~$n\in\textrm{\BBB Z}_\nu$. The
identity $\xi(c_1\varphi+c_2\psi)=c_1\xi(\varphi)+c_2\xi(\psi)$
also holds for all $\varphi,\psi\in\Phi$ and real numbers $c_1$
and~$c_2$.

Let us also define the function
$\varphi^{(N,A_N)}(x)=\frac1{A_N}\varphi(\frac xN)$ for all
functions $\varphi\in\Phi$ and positive integers $N=1,2,\dots$,
with some appropriately chosen constants $A_N>0$. Observe that
$\xi(\varphi^{(N,A_N)}_n)=Z^N_n$ with the random variable 
$Z^N_n$ defined in~(\ref{(1.1)}). All previously
introduced notions related to discrete random fields can be
reformulated with the help of the set of random variables
$\xi(\varphi)$, $\varphi\in\Phi$. Thus for instance the random
field $\xi_n$, $n\in\textrm{\BBB Z}_\nu$ is self-similar with
self-similarity parameter~$\alpha$ if and only if
$\xi(\varphi^{(N,N^\alpha)})\stackrel{\Delta}{=}\xi(\varphi)$
for all $\varphi\in\Phi$ and $N=1,2,\dots$. (To see why this
statement holds observe that the distributions of two random
vectors agree if and only if every linear combination of their
coordinates have the same distribution. This follows from the fact
that the characteristic function of a random vector determines its
distribution.)

It will be useful to define the continuous time version of
discrete random fields as generalized random fields. The generalized
random fields will be defined as a set of random variables indexed
by the elements of a linear space of functions. They show some
similarity to the class of random variables $\xi(\varphi)$,
$\varphi\in\Phi$, defined above. The main difference is that instead
of the space~$\Phi$ a different linear space is chosen for the
parameter set of the random field. We shall choose the so-called
Schwartz space for this role.

Let ${\cal S}={\cal S}_\nu$ be the Schwartz space of (real valued) 
rapidly decreasing, smooth functions on $R^\nu$. (See 
e.g.~\cite{r15} for the definition of ${\cal S}_\nu$. 
\index{space of test functions in the Schwartz space of 
generalized functions}
I shall present a more detailed discussion about the definition 
of the space ${\cal S}$ together with the topology introduced in
it in the adjustment to Chapter~1.)  Generally one takes the 
space of complex valued, rapidly decreasing, smooth functions 
as the  space~${\cal S}$, but we shall denote the space of 
{\it real valued},\/ rapidly decreasing, smooth functions 
by~${\cal S}$ if we do not say this otherwise. We shall omit
the subscript $\nu$ if it leads to no ambiguity. Now we 
introduce the notion of generalized random fields.

\medskip\noindent
{\bf Definition of Generalized Random Fields.} 
\index{generalized random field} 
{\it We say that the set of random variables $X(\varphi)$,  
$\varphi\in{\cal S}$, is a generalized random field over the 
Schwartz space ${\cal S}$ of rapidly decreasing, smooth functions if:

\medskip
\begin{description}
\item[(a)]  $X(a_1\varphi_1+a_2\varphi_2)=a_1X(\varphi_1)+a_2X(\varphi_2)$
with probability 1 for all real numbers $a_1$ and $a_2$ and
$\varphi_1\in{\cal S}$, $\varphi_2\in{\cal S}$. (The exceptional set of
probability~0 where this identity does not hold may depend on $a_1$,
$a_2$, $\varphi_1$ and $\varphi_2$.)
\item[(b)] $X(\varphi_n)\Rightarrow X(\varphi)$ stochastically if
$\varphi_n\to\varphi$ in the topology of ${\cal S}$.
\end{description}

}
\medskip
We also introduce the following definitions.

\medskip\noindent
{\bf Definition of Stationarity and Gaussian Property of a 
Generalized Random Field and the Notion of Convergence of 
Generalized Random Fields in Distribution.} 
\index{Gaussian random field (generalized field)}
\index{stationary random field (generalized field)}
\index{convergence of generalized random fields in distribution}
{\it The generalized random field
$X=\{X(\varphi),\,\varphi\in {\cal S}\}$ is stationary if
$X(\varphi)\stackrel{\Delta}{=}X(T_t\varphi)$ for all 
$\varphi\in{\cal S}$ and $t\in R^\nu$, where 
$T_t\varphi(x)=\varphi(x-t)$. It is Gaussian
if $X(\varphi)$ is a Gaussian random variable for all
$\varphi\in{\cal S}$. The relation
$X_n\stackrel{{\cal D}}{\rightarrow} X_0$ as $n\to\infty$ 
holds for a sequence of generalized random fields $X_n$, 
$n=0,1,2,\dots$, if 
$X_n(\varphi)\stackrel{{\cal D}}{\rightarrow} X_0(\varphi)$ 
for all $\varphi\in{\cal S}$, where
$\stackrel{{\cal D}}{\rightarrow}$ denotes convergence in 
distribution.}

\medskip
Given a stationary generalized  random field $X$ and a function
$A(t)>0$, $t>0$, on the set of positive real numbers we define the
(stationary) random fields $X^A_t$ for all $t>0$ by the formula
\begin{equation}
X^A_t(\varphi)=X(\varphi_t^A), \quad \varphi\in{\cal S}, \qquad
\textrm{where } \varphi_t^A(x)=A(t)^{-1}\varphi\left(\frac xt\right). 
\label{(1.3)}
\end{equation}

We are interested in the following

\medskip\noindent
{\bf Question.} {\it When does a generalized random field $X^*$ 
exist such that $X_t^A\stackrel{{\cal D}}{\rightarrow} X^*$ as 
$t\to\infty$ (or as $t\to0$)?}

\medskip
In relation to this question we introduce the following

\medskip\noindent
{\bf Definition of Self-similarity.} 
\index{self-similar random field (generalized field)}
\index{self-similarity parameter} 
{\it The stationary generalized random field $X$ is self-similar 
with self-similarity parameter $\alpha$ if  
$X^A_t(\varphi)\stackrel{\Delta}{=} X(\varphi)$ for all
$\varphi\in{\cal S}$  and $t>0$ with the function 
$A(t)=t^\alpha$.}

\medskip
To answer the above question one should first describe the 
generalized self-similar random fields.

We try to explain the motivation behind the above definitions. 
Given an ordinary random field $X(t)$, $t\in R^\nu$, and a 
topological space ${\cal E}$ consisting of functions over 
$R^\nu$ one can define the random variables 
$X(\varphi)=\int_{R^\nu} \varphi(t)X(t)\,dt$, 
$\varphi\in{\cal E}$. Some difficulty may arise when defining 
this integral, but it can be overcome in all interesting cases. 
If the space ${\cal E}$ is rich enough, and this is the case 
if ${\cal E}={\cal S}$, then the integrals $X(\varphi)$, 
$\varphi\in{\cal E}$, determine the random process $X(t)$. The 
set of random variables $X(\varphi)$, $\varphi\in{\cal E}$, is 
a generalized random field in all nice cases. On the other hand, 
there are generalized random fields which cannot be obtained by 
integrating ordinary random fields. In particular, the 
generalized self-similar random fields we shall construct later 
cannot be interpreted through ordinary fields. The above 
definitions of various properties of generalized fields are
fairly natural, considering what these definitions mean for
generalized random fields obtained by integrating ordinary fields.

The investigation of generalized random fields is simpler than that
of ordinary discrete random fields, because in the continuous case
more symmetry is available. Moreover, in the study or construction
of discrete random fields generalized random fields may play a useful
role. To understand this let us remark that if we have a generalized
random field $X(\varphi)$, $\varphi\in{\cal S}$, and we can extend the
space ${\cal S}$ containing the test function $\varphi$ to such a larger
linear space ${\cal T}$ for which $\Phi\subset{\cal T}$ with the above
introduced linear space~$\Phi$, then we can define the discrete random
field $X(\varphi)$, $\varphi\in\Phi$, by a restriction of the space of
test functions of the generalized random field $X(\varphi)$,
$\varphi\in{\cal T}$. This random field can be considered as the
discretization of the  original generalized random field
$X(\varphi)$, $\varphi\in{\cal S}$.

We finish this chapter by defining the generalized subordinated
random fields. Then we shall explain the basic results about
the Schwartz space ${\cal S}$ and generalized functions in
a separate sub chapter.

Let $X(\varphi)$, $\varphi\in{\cal S}$, be a generalized 
stationary Gaussian random field. The formula
$T_tX(\varphi))=X(T_t\varphi)$, $T_t\varphi(x)=\varphi(x-t)$,
 defines the shift transformation for all $t\in R^\nu$. Let
${\cal H}$ denote the real Hilbert space consisting of the
${\cal B}={\cal B}(X(\varphi),\;\varphi\in{\cal S})$ measurable 
random variables with finite second moment. The shift 
transformation can be extended to a group of unitary 
transformations over ${\cal H}$ similarly to the discrete case.
This will be explained in more detail in the next chapter.

\medskip\noindent
{\bf Definition of Generalized Random Fields Subordinated to a
Generalized Stationary Gaussian Random Field.} 
\index{random field (generalized) subordinated to a generalized
stationary Gaussian random field} 
{\it Given a generalized stationary Gaussian random field 
$X(\varphi)$, $\varphi\in{\cal S}$, we define the Hilbert 
space ${\cal H}$ and the shift transformations $T_t$, 
$t\in R^\nu$, over ${\cal H}$ as above. A generalized 
stationary random field $\xi(\varphi)$, $\varphi\in{\cal S}$, 
is subordinated to the field $X(\varphi)$,
$\varphi\in{\cal S}$, if $\xi(\varphi)\in{\cal H}$ and
$T_t\xi(\varphi)=\xi(T_t\varphi)$ for all $\varphi\in{\cal S}$ 
and $t\in R^\nu$, and $E[\xi\varphi_n)-\xi(\varphi)]^2\to0$ if
$\varphi_n\to\varphi$ in the topology of ${\cal S}$.}

\section{A Brief Overview About Some Results on 
Generalized Functions}

Let us first describe the Schwartz spaces ${\cal S}$ and 
${\cal S}^c$ in more detail. The space 
${\cal S}^c=({\cal S}_\nu)^c$ consists of those complex valued 
functions of $\nu$ variables which decrease at infinity, 
together with their derivatives, faster than any polynomial
degree. More explicitly, $\varphi\in{\cal S}^c$ for a 
complex valued function $\varphi$ of $\nu$ variables if
$$
\left|x_1^{k_1}\cdots x_\nu^{k_\nu}\frac{\partial^{q_1+\cdots+q_\nu}}
{\partial x_1^{q_1}\dots \partial x_\nu^{q_\nu}}
\varphi(x_1,\dots,x_\nu)\right|
\le C(k_1,\dots,k_\nu,q_1,\dots,q_\nu)
$$
for all point $x=(x_1,\dots,x_\nu)\in R^\nu$ and  vectors
$(k_1,\dots,k_\nu)$, $(q_1,\dots,q_\nu)$ with non-negative
integer coordinates with some constant
$C(k_1,\dots,k_\nu,q_1,\dots,q_\nu)$ which may depend on the
function~$\varphi$. This formula can be written in a more 
concise form as
$$
|x^k D^q\varphi(x)|\le C(k,q) \quad \textrm{with } 
k=(k_1,\dots,k_\nu)  \textrm{ and } q=(q_1,\dots,q_\nu),
$$
where $x=(x_1,\dots,x_\nu)$, $x^k=x_1^{k_1}\cdots x_\nu^{k_\nu}$ 
and $D^q=\frac{\partial^{q_1+\cdots+q_\nu}}
{\partial x_1^{q_1}\dots \partial x_\nu^{q_\nu}}$.
The elements of the space ${\cal S}$ are defined
similarly, with the only difference that they are real 
valued functions.

To define the spaces ${\cal S}$ and ${\cal S}^c$ we still have 
to define the convergence in them. We say that a sequence of 
functions $\varphi_n\in{\cal S}^c$ (or $\varphi_n\in{\cal S}$) 
converges to a function $\varphi$ if
$$
\lim_{n\to\infty}\sup_{x\in R^\nu}
(1+|x|^2)^k|D^q\varphi_n(x)-D^q\varphi(x)|=0.
$$
for all $k=1,2,\dots$ and $q=(q_1,\dots,q_\nu)$.
It can be seen that the limit function $\varphi$ is also in the
space~${\cal S}^c$ (or in the space ${\cal S}$).

A nice topology can be introduced in the space ${\cal S}^c$ (or 
${\cal S}$) which induces the above convergence. The following 
topology is an appropriate choice. 
\index{space of test functions in the Schwartz space of 
generalized functions}
Let a basis of neighbourhoods of the origin consist
of the sets
$$
U(k,q,\varepsilon)=\left\{\varphi\colon\;
\max_x(1+|x|^2)^k |D^q\varphi(x)|<\varepsilon\right\}
$$
with $k=0,1,2,\dots$, $q=(q_1,\dots,q_\nu)$ with non-negative 
integer coordinates and $\varepsilon>0$, where 
$|x|^2=x_1^2+\cdots+x_\nu^2$. A basis of neighbourhoods of an 
arbitrary function $\varphi\in{\cal S}^c$ (or $\varphi\in{\cal S}$) 
consists of sets of the form $\varphi+U(k,q,\varepsilon)$,
where the class of sets $U(k,q,\varepsilon)$ is a basis of 
neighbourhood of the origin. The fact that the convergence in 
${\cal S}$ has such a representation, (and a similar result 
holds in some other spaces studied in the theory of 
generalized functions) has a great importance in the theory 
of generalized functions. We also have exploited this fact 
in Chapter~6 of this Lecture Note. Topological
spaces with such a topology are called countably normed spaces.

The space of generalized functions ${\cal S}'$ consists of the
{\it continuous}\/ linear maps $F\colon\;{\cal S}\to C$ or
$F\colon\;{\cal S}^c\to C$, where $C$ denotes the linear space of
complex numbers.  (In the study of the space ${\cal S}'$ we omit
the upper index~$c$, i.e. we do not indicate whether we are working in
real or complex space when this causes no problem.) We shall write the
map $F(\varphi)$, $F\in{\cal S}'$ and $\varphi \in{\cal S}$ (or
$\varphi\in{\cal S}^c$) in the form~$(F,\varphi)$.
\index{Schwartz space of generalized functions.}

We can define generalized functions $F\in{\cal S}'$ by the formula
$$
(F,\varphi)=\int \overline{f(x)}\varphi(x)\,dx \quad\textrm{for all }
\varphi\in{\cal S} \quad\textrm{ or  }\varphi\in{\cal S}^c
$$ 
with a function $f$ such
that $\int(1+|x|^2)^{-p}|f(x)|\,dx<\infty$ with some $p\ge0$.
(The upper script \ $\bar{ }$ \ denotes complex conjugate in 
the sequel.) Such functionals are called regular. There are 
also non-regular functionals in the space~${\cal S}'$. An 
example for them is the $\delta$-function defined by the 
formula $(\delta ,\varphi)=\varphi(0)$. There is a good 
description of the generalized functions $F\in{\cal S}'$, (see 
the book I.~M.~Gelfand and G. E.~Shilov: Generalized functions, 
Volume~2, Chapter~2, Chapter~4), but we do not need this result, 
hence we do not discuss it here. Another important question in 
this field not discussed in the present note is about the 
interpretation of a usual function as a generalized function 
in the case when it does not define a regular function in 
${\cal S}'$ because of its strong singularity in some points. 
In such cases some regularization can be applied. It is  an 
important problem in the theory of generalized functions to 
define the appropriate generalized functions in such cases, 
but it does not appear in the study of the problems in this 
work.

The derivative and the Fourier transform of generalized functions are
also defined, and they play an important role in some investigations.
In the definition of these notions for generalized functions we want
to preserve the old definition if nice regular functionals are
considered for which these notions were already defined in classical
analysis. Such considerations lead to the definition
$(\frac{\partial_j}{\partial x_j}F,\varphi)
=-(F,\frac{\partial\varphi}{\partial x_j})$  of the derivative of
generalized functions. We do not discuss this definition in more
detail, because here we do not work with the derivatives of
generalized functions.

The Fourier transform of generalized functions in~$S'$ appears in
our discussion, although only in an implicit form. The
Bochner-Schwartz theorem discussed in Chapter~3 actually deals with
the Fourier transform of generalized functions. Hence the definition
of Fourier transform will be given in more detail.

We shall define the Fourier transform of a generalized function by
means of a natural extension of the Parseval formula, more explicitly
of a simplified version of it, where the same identity
$$
\int_{R^\nu} \overline{f(x)}g(x)\,dx
=\frac1{(2\pi)^\nu}\int_{R^\nu} \overline{\tilde f(u)}\tilde g(u)\,du
$$
is formulated with $\tilde f(u)=\int_{R^\nu}e^{i(u,x)}f(x)\,dx$
and $\tilde g(u)=\int_{R^\nu}e^{i(u,x)}g(x)\,dx$. But now we
consider a pair of functions $(f,g)$ with different properties.
We demand that $f$ should be an integrable function, and
$g\in{\cal S}^c$. (In the original version of the Parseval formula
both~$f$ and~$g$ are $L_2$ functions.)

The proof of this identity is simple. Indeed, since the function
$g\in{\cal S}^c$ can be calculated as the inverse Fourier transform
of its Fourier transform~$\tilde g\in{\cal S}^c$, i.e.\
$g(x)=\frac1{(2\pi)^\nu}\int e^{-i(u,x)}\tilde g(u)\,du$, we can
write
\begin{eqnarray*}
\int \overline{f(x)}g(x)\,dx
&=&\int \overline{f(x)}\left[\frac1{(2\pi)^\nu}\int e^{-i(u,x)}
\tilde g(u)\,du\right]\,dx\\
&=&\int\tilde g(u)\left[\frac1{(2\pi)^\nu}
\int\overline{ e^{i(u,x)}f(x)}\,dx\right]\,du \\
&=&\frac1{(2\pi)^\nu}\int \overline{\tilde f(u)}\tilde g(u)\,du.
\end{eqnarray*}
\index{Fourier transform of a generalized function} 
Let us also remark that the Fourier transform $f\to\tilde f$ is a
bicontinuous map from ${\cal S}^c$ to~${\cal S}^c$. (This means 
that this transformation is invertible, and both the Fourier 
transform and its inverse are continuous maps from ${\cal S}^c$ 
to ${\cal S}^c$.) (The restriction of the Fourier transform to 
the space ${\cal S}$ of real valued functions is a bicontinuous 
map from ${\cal S}$ to the subspace of ${\cal S}^c$ consisting 
of those functions $f\in{\cal S}^c$ for which
$f(-x)=\overline{f(x)}$ for all $x\in R^\nu$.)

The above results make natural the following definition of the Fourier
transform~$\tilde F$ of a generalized function $F\in{\cal S}'$.
$$
(\tilde F, \tilde\varphi)=(2\pi)^\nu(F,\varphi)
\quad\textrm{for all } \varphi\in{\cal S}^c.
$$
Indeed, if $F\in{\cal S}'$ then $\tilde F$ is also a continuous 
linear map on ${\cal S}^c$, i.e. it is also an element 
of~${\cal S}'$. Besides, the above proved version of the Parseval 
formula implies that if we consider an integrable function~$f$ 
on~$R^\nu$ both as a usual function and as a (regular) 
generalized function, its Fourier transform agrees in the two cases.

\medskip
There are other classes of test functions and spaces of generalized
functions studied in the literature. The most popular among them 
is the space~${\cal D}$ of infinitely many times differentiable 
functions with compact support and its dual space~${\cal D}'$, 
the space of continuous linear transformations on the space 
${\cal D}$. 
\index{generalized functions with the test function space of smooth
functions with compact support}
(These spaces are
generally denoted  by~${\cal D}$ and~${\cal D}'$ in the literature,
although just the book~\cite{r15} that we use as our main reference
in this subject applies the notation ${\cal K}$ and~${\cal K}'$ for
them.) We shall discuss this space only very briefly.

The space ${\cal D}$ consists of the infinitely many times
differentiable functions with compact support. Thus it is a
subspace of~${\cal S}$. A sequence $\varphi_n\in{\cal D}$, $n=1,2,\dots$,
converges to a function~$\varphi$, if there is a compact set
$A\subset R^\nu$ which is the support of all these
functions~$\varphi_n$, and the functions $\varphi_n$ together with
all their derivatives converge uniformly to the
function~$\varphi$ and to its corresponding derivatives. It is not
difficult to see that also $\varphi\in{\cal D}$, and if the functions
$\varphi_n$ converge to $\varphi$ in the space~${\cal D}$, then they
also converge to~$\varphi$ in the space~${\cal S}$. Moreover, ${\cal D}$
is an everywhere dense subspace of~${\cal S}$. The space
${\cal D}'$ consists of the continuous linear functionals in~${\cal D}$.

The results describing the behaviour of ${\cal D}$ and ${\cal D}'$ are
very similar to those describing the behaviour of ${\cal S}$
and~${\cal S}'$. There is one difference that deserves some attention.
The Fourier transforms of the functions in ${\cal D}$ may not belong
to~${\cal D}$. The class of these Fourier transforms can be described
by means of some results in complex analysis. A topological
space~${\cal Z}$ can be defined on the set of Fourier transforms of
the functions from the space~${\cal D}$. If we want to apply Fourier
analysis in the space~${\cal D}$, then we also have to study this
space~${\cal Z}$ and its dual space~${\cal Z}'$. I omit the details.

\chapter{Wick Polynomials}

In this chapter we consider the so-called Wick polynomials, a
multi-dimensional generalization of Hermite polynomials. They are
closely related to multiple Wiener--It\^o integrals.

Let $X_t$, $t\in T$, be a set of jointly Gaussian random 
variables indexed by a parameter set $T$. Let $EX_t=0$ for all 
$t\in T$. We define the real Hilbert spaces ${\cal H}_1$ and 
${\cal H}$ in the following way: A square integrable random 
variable is in ${\cal H}$ if and only if it is measurable with 
respect to the $\sigma$-algebra ${\cal B}={\cal B}(X_t,\;t\in T)$, 
and the scalar product in ${\cal H}$ is defined as 
$(\xi,\eta)=E\xi\eta$, $\xi,\,\eta\in{\cal H}$. The Hilbert 
space ${\cal H}_1\subset{\cal H}$ is the subspace of ${\cal H}$ 
generated by the finite linear combinations $\sum c_jX_{t_j}$, 
$t_j\in T$. We consider only such sets of Gaussian random 
variables $X_t$ for which ${\cal H}_1$ is separable. Otherwise 
$X_t$, $t\in T$, can be arbitrary, but the most interesting case 
for us is when $T={\cal S}_\nu$ or $\textrm{\BBB Z}_\nu$, and 
$X_t$, $t\in T$, is a stationary Gaussian field.

Let $Y_1,Y_2,\dots$ be an orthonormal basis in ${\cal H}_1$. The
uncorrelated random variables $Y_1,Y_2,\dots$ are independent, since
they are (jointly) Gaussian. Moreover,
$$
{\cal B}(Y_1,Y_2,\dots)={\cal B}(X_t,\;t\in T).
$$  
Let $H_n(x)$ denote the $n$-th Hermite polynomial with leading 
coefficient~1, i.e. let 
$H_n(x)=(-1)^ne^{x^2/2}\frac{d^n}{dx^n}(e^{-x^2/2})$. 
\index{Hermite polynomials}
We recall the following results from analysis and measure theory.

\medskip\noindent
{\bf Theorem 2A.} {\it The Hermite polynomials $H_n(x)$, $n=0,1,2,\dots$,
form a complete orthogonal system in
$L_2\left(R,{\cal B},\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx\right)$.
(Here ${\cal B}$
denotes the Borel $\sigma$-algebra on the real line.)}

\medskip
Let  $(X_j,{\cal X}_j,\mu_j)$, $j=1,2,\dots$, be countably many 
independent copies of a probability space $(X,{\cal X},\mu)$. 
(We denote the points of $X_j$ by $x_j$.) Let 
$(X^\infty,{\cal X}^\infty,\mu^\infty)
=\prod\limits_{j=1}^\infty(X_j,{\cal X}_j,\mu_j)$. With such a 
notation the following result holds.

\medskip\noindent
{\bf Theorem 2B.} {\it Let $\varphi_0,\varphi_1,\dots$,
$\varphi_0(x)\equiv1$, be a complete orthonormal system in the
Hilbert space $L_2(X,{\cal X},\mu)$. Then the functions
$\prod\limits_{j=1}^\infty\varphi_{k_j}(x_j)$, where only 
finitely many indices $k_j$ differ from~0, form a complete 
orthonormal basis in $L_2(X^\infty,{\cal X}^\infty,\mu^\infty)$.}

\medskip\noindent
{\bf Theorem 2C.} {\it Let $Y_1,Y_2,\dots$ be random variables 
on a probability space $(\Omega,{\cal A},P)$ taking values in 
a measurable space  $(X,{\cal X})$. Let $\xi$ be a real valued 
random variable measurable with respect to the $\sigma$-algebra 
${\cal B}(Y_1,Y_2,\dots)$, and let $(X^\infty,{\cal X}^\infty)$ 
denote the infinite product
$(X\times X\times\cdots,{\cal X}\times {\cal X}\times\cdots)$ 
of the space $(X,{\cal X})$ with itself. Then there exists a 
real valued, measurable function~$f$ on the space 
$(X^\infty,{\cal X}^\infty)$ such that $\xi=f(Y_1,Y_2,\dots)$.}

\medskip\noindent
{\it Remark.}\/ Let us have a stationary random field 
$X_n(\omega)$, $n\in\textrm{\BBB Z}_\nu$. Theorem~2C enables 
us to extend the shift transformation $T_m$, defined as 
$T_mX_n(\omega)=X_{n+m}(\omega)$, $n,\,m\in\textrm{\BBB Z}_\nu$, 
for all random variables $\xi(\omega)$, measurable with respect 
to the $\sigma$-algebra
${\cal B}(X_n(\omega),\,n\in\textrm{\BBB Z}_\nu)$. Indeed, by
Theorem~2C we can write 
$\xi(\omega)=f(X_n(\omega),\,n\in\textrm{\BBB Z}_\nu)$,
and define 
$T_m\xi(\omega)=f(X_{n+m}(\omega),\,n\in\textrm{\BBB Z}_\nu)$. 
We still have to understand, that although the function~$f$ is 
not unique in the representation of the random 
variable~$\xi(\omega)$, the above definition of 
$T_m\xi(\omega)$ is meaningful. To see this we have to 
observe that if $f_1(X_n(\omega),\,n\in\textrm{\BBB Z}_\nu)
=f_2(X_n(\omega),\,n\in\textrm{\BBB Z}_\nu)$ 
for two functions $f_1$ and $f_2$ with probability~1, then 
also $f_1(X_{n+m}(\omega),\,n\in\textrm{\BBB Z}_\nu)
=f_2(X_{n+m}(\omega),\,n\in\textrm{\BBB Z}_\nu)$
with probability~1 because of the stationarity of the random
field~$X_n(\omega)$, $n\in\textrm{\BBB Z}_\nu$. Let us also 
observe that $\xi(\omega)\stackrel{\Delta}{=}T_m\xi(\omega)$ 
for all $m\in\textrm{\BBB Z}_\nu$. Besides, $T_m$ is a linear 
operator on the linear space of random variables, measurable 
with respect to the $\sigma$-algebras
${\cal B}(X_n,\,n\in\textrm{\BBB Z}_\nu)$. If we restrict it 
to the space of square integrable random variables, then 
$T_m$ is a unitary operator, and the operators $T_m$, 
$m\in\textrm{\BBB Z}_\nu$, constitute a unitary
group.
\index{shift transformation (discrete random field)}

Let a stationary generalized random field 
$X=\{X(\varphi),\,\varphi\in{\cal S}\}$ be given. The 
shift  $T_t\xi$ of a random variable $\xi$, measurable 
with respect to the $\sigma$-algebra 
${\cal B}(X(\varphi),\,\varphi\in{\cal S})$ can be 
defined  for all $t\in R^\nu$ similarly to the discrete 
case with  the help of Theorem~2C and the following 
result. If $\xi\in{\cal B}(X(\varphi),\,\varphi\in{\cal S})$ 
for a random variable~$\xi$, then there exists such a 
countable subset $\{\varphi_1,\varphi_2,\dots\}\subset{\cal S}$ 
(depending on the random variable~$\xi$) for which $\xi$ is
${\cal B}(X(\varphi_1),X(\varphi_2),\dots)$ measurable. (We 
write 
$\xi(\omega)=f(X(\varphi_1)(\omega),X(\varphi_2)(\omega),\dots)$ 
with appropriate functions $f$, and $\varphi_1\in{\cal S}$,
$\varphi_2\in{\cal S}$,\dots, and define the shift $T_t\xi$ as
$T_t\xi(\omega)
=f(X(T_t\varphi_1)(\omega),X(T_t\varphi_2)(\omega),\dots)$,
where $T_t\varphi(x)=\varphi(x-t)$ for $\varphi\in{\cal S}$.) The
transformations $T_t$, $t\in R^{\nu}$, are linear operators 
over the space of random variables measurable with respect to the
$\sigma$-algebra ${\cal B}(X(\varphi),\,\varphi\in{\cal S})$ 
with similar properties as their discrete counterpart.
\index{shift transformation (generalized random field)}

\medskip
Theorems 2A, 2B and 2C have the following important consequence.

\medskip\noindent
{\bf Theorem 2.1.} {\it Let $Y_1,Y_2,\dots$ be an orthonormal basis
in the Hilbert space ${\cal H}_1$ defined above with the help of a set
of Gaussian random variables $X_t$, $t\in T$. Then the set of all
possible finite products $H_{j_1}(Y_{l_1})\cdots H_{j_k}(Y_{l_k})$
is a complete orthogonal system in the Hilbert space ${\cal H}$
defined above. (Here $H_j(\cdot)$ denotes the $j$-th Hermite
polynomial.)}

\medskip\noindent
{\it Proof of Theorem 2.1.}\/ By Theorems~2A and~2B the set of
all possible products $\prod\limits_{j=1}^\infty H_{k_j}(x_j)$, 
where only finitely many indices $k_j$ differ from 0, is a 
complete orthonormal system in 
$L_2\left(R^\infty,{\cal B}^\infty,\prod\limits_{j=1}^\infty
\frac{e^{-x_j^2/2}}{\sqrt{2\pi}}\,dx_j\right)$. Since
${\cal B}(X_t,\;t\in T)={\cal B}(Y_1,Y_2,\dots)$, Theorem~2C 
implies that the mapping $f(x_1,x_2,\dots,)\to f(Y_1,Y_2,\dots)$ 
is a unitary transformation from 
$L_2\left(R^\infty,{\cal B}^\infty,\prod\limits_{j=1}^\infty
\frac{e^{-x_j^2/2}}{\sqrt{2\pi}}\,dx_j\right)$ to ${\cal H}$. 
(We call a transformation from a Hilbert space to another Hilbert 
space unitary if it is norm preserving and invertible.) Since the 
image of a complete orthogonal system under a unitary 
transformation is again a complete orthogonal system, 
Theorem~2.1 is proved.  \hfill$\qed$

\medskip
Let ${\cal H}_{\le n}\subset{\cal H}$, $n=1,2,\dots$, (with the 
previously introduced Hilbert space ${\cal H}$) denote the Hilbert 
space which is the closure of the linear space consisting of the 
elements $P_n(X_{t_1},\dots,X_{t_m})$, where $P_n$ runs through 
all polynomials of degree less than or equal to~$n$, and the 
integer~$m$ and indices $t_1,\dots,t_m\in T$ are arbitrary. Let 
${\cal H}_0={\cal H}_{\le 0}$ consist of the constant functions, 
and let ${\cal H}_n={\cal H}_{\le n}\ominus{\cal H}_{\le n-1}$, 
$n=1,2,\dots$, where $\ominus$ denotes orthogonal completion. It 
is clear that the Hilbert space ${\cal H}_1$ given in this 
definition agrees with the previously defined Hilbert space 
${\cal H}_1$. If $\xi_1,\dots,\xi_m\in{\cal H}_1$, and 
$P_n(x_1,\dots,x_m)$ is a polynomial of degree $n$, then
$P_n(\xi_1,\dots,\xi_m)\in{\cal H}_{\le n}$. Hence Theorem~2.1 
implies that
\begin{equation}
{\cal H}={\cal H}_0+{\cal H}_1+{\cal H}_2+\cdots,  \label{(2.1)}
\end{equation}
where $+$ denotes direct sum. Now we introduce the following

\medskip\noindent
{\bf Definition of Wick Polynomials.} 
\index{Wick polynomials} 
{\it Given a polynomial $P(x_1,\dots,x_m)$ of degree~$n$ and 
a set of (jointly) Gaussian random variables 
$\xi_1,\dots,\xi_m\in{\cal H}_1$, the Wick polynomial
\hbox{$\colon\!P(\xi_1,\dots,\xi_m)\!\colon$} is the orthogonal 
projection of the random variable $P(\xi_1,\dots,\xi_m)$ to the 
above defined subspace ${\cal H}_n$ of the Hilbert 
space ${\cal H}$.}

\medskip
It is clear that Wick polynomials of different degree are orthogonal.
Given some $\xi_1,\dots,\xi_m\in{\cal H}_1$ define the subspaces
${\cal H}_{\le n}(\xi_1,\dots,\xi_m)\subset{\cal H}_{\le n}$, $n=1,2,\dots$,
as the set of all polynomials of the random variables 
$\xi_1,\dots,\xi_m$ with degree less than or equal to~$n$. Let
${\cal H}_{\le0}(\xi_1,\dots,\xi_m)
={\cal H}_{0}(\xi_1,\dots,\xi_m)={\cal H}_0$,
and ${\cal H}_n(\xi_1,\dots,\xi_m)={\cal H}_{\le n}(\xi_1,\dots,\xi_m)\ominus
{\cal H}_{\le n-1}(\xi_1,\dots,\xi_m)$. With the help of 
this notation we formulate the following

\medskip\noindent
{\bf Proposition 2.2.} {\it Let $P(x_1,\dots,x_m)$ be a polynomial 
of degree~$n$. Then the random polynomial
$\colon\!P(\xi_1,\dots,\xi_m)\!\colon$ equals the orthogonal
projection of $P(\xi_1,\dots,\xi_m)$ to ${\cal H}_n(\xi_1,\dots,\xi_m)$.}

\medskip\noindent
{\it Proof of Proposition 2.2.}\/ Let
$\colon\!\bar P(\xi_1,\dots,\xi_m)\!\colon$
denote the projection of the random polynomial 
$P(\xi_1,\dots,\xi_m)$ to ${\cal H}_n(\xi_1,\dots,\xi_m)$. Obviously
$$
P(\xi_1,\dots,\xi_m)-\colon\!\bar P(\xi_1,\dots,\xi_m)\!\colon\in
{\cal H}_{\le n-1}(\xi_1,\dots,\xi_m)\subseteq {\cal H}_{\le n-1}.
$$
Hence in order to prove Proposition~2.2 it is enough to show that for all
$\eta\in{\cal H}_{\le n-1}$
\begin{equation}
E\colon\!\bar P(\xi_1,\dots,\xi_m)\!\colon\eta=0, \label{(2.2)}
\end{equation}
since this means that $\colon\!\bar P(\xi_1,\dots,\xi_m)\!\colon$ is the
orthogonal projection of $P(\xi_1,\dots,\xi_m)\in{\cal H}_{\le n}$
to~${\cal H}_{\le n-1}$.

Let $\varepsilon_1,\varepsilon_2,\dots$ be an orthonormal 
system in ${\cal H}_1$, also
orthonormal to $\xi_1,\dots,\xi_m$, and such that
$\xi_1,\dots,\xi_m,\varepsilon_1,\varepsilon_2,\dots$ form 
a basis in ${\cal H}_1$.
If $\eta=\prod\limits_{i=1}^m\xi_i^{l_i}
\prod\limits_{j=1}^\infty\varepsilon_j^{k_j}$ with
such exponents $l_i$ and $k_j$ that $\sum l_i+\sum k_j\le n-1$, 
then (\ref{(2.2)}) holds for this random variable $\eta$ 
because of the independence of the random variables $\xi_i$ 
and $\varepsilon_j$. Since the linear combinations of such 
$\eta$ are dense in ${\cal H}_{\le n-1}$,
formula~(\ref{(2.2)}) and Proposition~(2.2) are proved.
\hfill$\qed$

\medskip\noindent
{\bf Corollary 2.3.} {\it Let $\xi_1,\dots,\xi_m$ be an orthonormal
system in ${\cal H}_1$, and let
$$
P(x_1,\dots,x_m)=\sum c_{j_1,\dots,j_m}x^{j_1}\cdots x_m^{j_m}
$$
be a homogeneous polynomial, i.e. let $j_1+\cdots j_m=n$ with some
fixed number~$n$ for all sets $(j_1,\dots,j_m)$ appearing in this
summation. Then
$$
\colon\!P(\xi_1,\dots,\xi_m)\!\colon=\sum
c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m).
$$
In particular,
$$
\colon\!\xi^n\!\colon=H_n(\xi) \quad \textrm{if } \xi\in{\cal H}_1,
\textrm{ and } E\xi^2=1.
$$
}

\medskip\noindent
{\it Remark.} Although we have defined the Wick polynomial (of degree~$n$)
for all polynomials $P(\xi_1,\dots,\xi_m)$ of degree~$n$, we could
have restricted our attention only to homogeneous polynomials of degree
$n$, since the contribution of each terms
$c(j_1,\dots j_m)\xi^{l_1}_1\cdots \xi_m^{l_m}$ of the polynomial
$P(\xi_1,\dots,\xi_m)$ such that $l_1+\cdots+l_m<n$ has a zero
contribution in the definition of the Wick polynomial
$\colon\!P(\xi_1,\dots,\xi_m)\!\colon$.

\medskip\noindent
{\it Proof of Corollary 2.3.}\/ Let the degree of the polynomial $P$
be~$n$. Then
\begin{equation}
P(\xi_1,\dots,\xi_m)
-\sum c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)
\in {\cal H}_{\le n-1}(\xi_1,\dots,\xi_m), \label{(2.3)}
\end{equation}
since $P(\xi_1,\dots,x_m)
-\sum c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)$
is a polynomial whose degree is less than~$n$. Let
$\eta=\xi_1^{l_1}\cdots\xi_m^{l_m}$, $\sum\limits_{i=1}^ml_i\le n-1$. Then
$$
E\eta H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)
=\prod_{i=1}^m E\xi_i^{l_i}H_{j_i}(\xi_i)=0,
$$
since $l_i<j_i$ for at least one index~$i$. Therefore
\begin{equation}
E\eta\sum c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)=0. 
\label{(2.4)}
\end{equation}
Since every element of ${\cal H}_{\le n-1}(\xi_1,\dots,\xi_m)$ 
can be written as the sum of such elements $\eta$, 
relation~(\ref{(2.4)}) holds for all 
$\eta\in{\cal H}_{\le n-1}(\xi_1,\dots,\xi_m)$. 
Relations~(\ref{(2.3)}) and~(\ref{(2.4)}) imply Corollary~2.3.
\hfill$\qed$

\medskip
The following statement is a simple consequence of the 
previous results.

\medskip\noindent
{\bf Corollary 2.4.} {\it Let $\xi_1,\xi_2,\dots$ be an 
orthonormal basis in ${\cal H}_1$. Then the random variables
$H_{j_1}(\xi_1)\cdots H_{j_k}(\xi_k)$, $k=1,2,\dots$,
$j_1+\cdots+j_k=n$, form a complete orthogonal basis in 
${\cal H}_n$.}

\medskip\noindent
{\it Proof of Corollary 2.4.} It follows from Corollary~2.3 that
$$
H_{j_1}(\xi_1)\cdots H_{j_k}(\xi_k)
=\colon\!\xi_1^{j_1}\cdots\xi_k^{j_k}\!\colon\in
{\cal H}_n\quad \textrm{for all } k=1,2,\dots
$$
if $j_1+\cdots+j_k=n$.
These random variables are orthogonal, and all Wick polynomials
$\colon\!P(\xi_1,\dots,\xi_m)\!\colon$ of degree~$n$ of the 
random variables $\xi_1,\xi_2,\dots$ can be represented as 
the linear combination of such terms. Since these Wick 
polynomials are dense in ${\cal H}_n$, this implies 
Corollary~2.4. \hfill$\qed$

\medskip
The arguments of this chapter exploited heavily some properties 
of Gaussian random variables. Namely, they exploited that the 
linear combinations of Gaussian random variables are again 
Gaussian, and in Gaussian case orthogonality implies 
independence. This means in particular, that the rotation 
of a standard normal vector leaves its distribution invariant. 
We finish this chapter with an observation based on these 
facts. This may illuminate the content of formula~(\ref{(2.1)}) 
from another point of view. We shall not use the results of 
the subsequent considerations in the rest of this work.

Let $U$ be a unitary transformation over ${\cal H}_1$. It can 
be extended to a unitary transformation ${\cal U}$ over 
${\cal H}$ in a natural way. Fix an orthonormal basis 
$\xi_1,\xi_2,\dots$ in ${\cal H}_1$, and define
${\cal U}1=1$,
${\cal U}\xi_{j_1}^{l_1}\cdots\xi_{j_k}^{l_k}=(U\xi_{j_1})^{l_1}\cdots
(U\xi_{j_k})^{l_k}$. This transformation can be extended to a 
linear transformation ${\cal U}$ over ${\cal H}$ in a unique 
way. The transformation ${\cal U}$ is norm preserving, since 
the joint distributions of $(\xi_{j_1},\xi_{j_2},\dots)$ and 
$(U\xi_{j_1},U\xi_{j_2},\dots)$ coincide. Moreover, it is unitary, 
since $U\xi_1,U\xi_2,\dots$ is an orthonormal basis in 
${\cal H}_1$. It is not difficult to see that if
$P(x_1,\dots,x_m)$ is an arbitrary polynomial, and
$\eta_1,\eta_2\dots,\eta_m\in{\cal H}_1$, then
${\cal U}P(\eta_1,\dots,\eta_m)=P(U\eta_1,\dots,U\eta_m)$. 
This relation means in particular that the transformation 
${\cal U}$ does not depend on the choice of the basis in 
${\cal H}_1$. If the transformations ${\cal U}_1$ and 
${\cal U}_2$ correspond to two unitary transformations $U_1$ 
and $U_2$ on ${\cal H}_1$, then the transformation
${\cal U}_1{\cal U}_2$ corresponds to $U_1U_2$. The subspaces
${\cal H}_{\le n}$ and therefore the subspaces ${\cal H}_n$ 
remain invariant under the transformations ${\cal U}$.

The shift transformations of a stationary Gaussian field, and 
their extensions to ${\cal H}$ are the most interesting 
examples for such unitary transformations $U$ and ${\cal U}$. 
In the terminology of group representations the above facts 
can be formulated in the following way: The mapping 
$U\to{\cal U}$ is a group representation  of $U({\cal H}_1)$
over ${\cal H}$, where $U({\cal H}_1)$ denotes the group of 
unitary transformations over ${\cal H}_1$. Formula~(\ref{(2.1)}) 
gives a decomposition of ${\cal H}$ into orthogonal invariant 
subspaces of this representation.

\chapter{Random Spectral Measures}

Some standard theorems of probability theory state that the correlation
function of a stationary random field can be expressed as the Fourier
transform of a so-called spectral measure. In this chapter we construct
a random measure with the help of these results, and express the
random field itself as the Fourier transform of this random measure in
some sense. We restrict ourselves to the Gaussian case, although most
of the results in this chapter are valid for arbitrary stationary
random field with finite second moment if independence is replaced by
orthogonality. In the next chapter we define the multiple Wiener--It\^o
integrals with respect to this random measure. In the definition of
multiple stochastic integrals the Gaussian property will be heavily
exploited. First we recall two results about the spectral representation
of the covariance function.

Given a stationary Gaussian field $X_n$, $n\in\textrm{\BBB Z}_\nu$, or
$X(\varphi)$, $\varphi\in{\cal S}$, we shall assume throughout the
paper that $EX_n=0$, $EX_n^2=1$ in the discrete and $EX(\varphi)=0$
in the generalized field case.

\medskip\noindent
{\bf Theorem 3A. (Bochner.)}
\index{Bochner theorem} 
{\it Let $X_n$, $n\in\textrm{\BBB Z}_\nu$, be a discrete 
(Gaussian) stationary random field. There exists a unique
probability  measure $G$ on $[-\pi,\pi)^\nu$ such that the 
correlation function $r(n)=EX_0X_n=EX_kX_{k+n}$, 
$n\in\textrm{\BBB Z}_\nu$, $k\in \textrm{\BBB Z}_\nu$, can be 
written in the form
\begin{equation}
r(n)=\int e^{i(n,x)}G(\,dx), \label{(3.1)}
\end{equation}
where $(\cdot,\cdot)$ denotes scalar product. Further 
$G(A)=G(-A)$ for all $A\in [-\pi,\pi)^\nu$.}

\medskip
We can identify $[-\pi,\pi)^\nu$ with the torus
$R^\nu/2\pi\textrm{\BBB Z}_\nu$. Thus
e.g. $-(-\pi,\dots,-\pi)=(-\pi,\dots,-\pi)$.

\medskip\noindent
{\bf Theorem 3B. (Bochner--Schwartz.)} 
\index{Bochner--Schwartz theorem}
{\it Let $X(\varphi)$, $\varphi\in{\cal S}$, be a generalized 
Gaussian stationary random field over ${\cal S}={\cal S}_\nu$. 
There exists a unique $\sigma$-finite measure $G$ 
on $R^\nu$ such that
\begin{equation}
EX(\varphi)X(\psi)=\int\tilde\varphi(x)\,\bar{\!\tilde\psi}(x)G(\,dx)
\qquad \textrm{for all } \varphi,\,\psi\in{\cal S}, \label{(3.2)}
\end{equation}
where $\,\tilde{}\,$ denotes Fourier transform and $\,\bar{}\,$
complex conjugate. The measure $G$ has
the properties $G(A)=G(-A)$ for all $A\in{\cal B}^\nu$, and
\begin{equation}
\int (1+|x|)^{-r}G(\,dx)<\infty \quad\textrm{with an appropriate } r>0.
\label{(3.3)}
\end{equation}
}

\medskip\noindent
{\it Remark.}\/ The above formulated results are actually not the
Bochner and Bochner--Schwartz theorem in their original form, they
are their consequences. In an Adjustment to Chapter~3 I formulate
the classical form of these theorems, and explain how the above
results follow from them.

\medskip
The measure $G$ appearing in Theorems~3A and~3B is called the spectral
measure of the stationary field. A measure~$G$ with the same
properties as the measure~$G$ in Theorem~3A or~3B will also be called
a spectral measure. This terminology is justified, since there exists
a stationary random field with spectral measure~$G$ for all such~$G$.
\index{spectral measure of a stationary random field 
(for discrete and generalized random fields)}

Let us now consider a stationary Gaussian random field 
(discrete or generalized one) with spectral measure $G$. We 
shall denote the space $L_2([-\pi,\pi)^\nu,{\cal B}^\nu,G)$ 
or $L_2(R^\nu,{\cal B}^\nu,G)$ simply by $L^2_G$. Let 
${\cal H}_1$ denote the real Hilbert space defined by means of 
the stationary random field, as it was done in Chapter~2. Let
${\cal H}^c_1$ denote its complexification, i.e. the elements of
${\cal H}^c_1$ are of the form $X+iY$, \ $X,\,Y\in{\cal H}_1$, 
and the scalar product is defined as
$(X_1+iY_1,X_2+iY_2)=EX_1X_2+EY_1Y_2+i(EY_1X_2-EX_1Y_2)$.
We are going to construct a unitary transformation $I$ 
from $L^2_G$ to ${\cal H}^c_1$. We shall define the random 
spectral measure via this transformation.

Let ${\cal S}^c$ denote the Schwartz space of rapidly decreasing, 
smooth, complex valued functions with the usual topology of the
 Schwartz space. (The elements of ${\cal S}^c$ are of the form 
$\varphi+i\psi$, \ $\varphi,\,\psi\in{\cal S}$.) We make the 
following observation. The finite linear combinations
$\sum c_ne^{i(n,x)}$ are dense in $L_G^2$ in the discrete field, 
and the functions $\varphi\in{\cal S}^c$ are dense in $L_G^2$ in 
the generalized field case. In the discrete field case this 
follows from the Weierstrass approximation theorem, which states 
that all continuous functions on $[-\pi,\pi)^\nu$ can be 
approximated arbitrary well in the supremum norm by 
trigonometrical polynomials. In the generalized field case let 
us first observe that the continuous functions with compact 
support are dense in $L^2_G$. We claim that also the functions 
of the space ${\cal D}$ are dense in $L^2_G$, where ${\cal D}$ 
denotes  the class of (complex valued) infinitely many times 
differentiable functions with compact support. Indeed, if 
$\varphi\in{\cal D}$ is real valued, $\varphi(x)\ge0$ for all
$x\in R^\nu$, $\int\varphi(x)\,dx=1$, we define
$\varphi_t(x)=t^\nu\varphi\left(\frac xt\right)$, and $f$ is a
continuous function with compact support, then 
$f*\varphi_t\to f$ uniformly as $t\to\infty$. Here $*$ denotes 
convolution. On the other hand, $f*\varphi_t\in{\cal D}$ for 
all $t>0$. Hence ${\cal D}\subset{\cal S}^c$ is dense in $L^2_G$.

Finally we recall the following result from the theory of 
distributions. The mapping $\varphi\to\tilde\varphi$ is an 
invertible, bicontinuous transformation from ${\cal S}^c$ 
into ${\cal S}^c$. In particular, the set of functions 
$\tilde\varphi$, \ $\varphi\in{\cal S}$, is also dense
in $L^2_G$.

Now we define the mapping
\begin{equation}
I\left(\sum c_n e^{i(n,x)}\right)=\sum c_nX_n \label{(3.4)}
\end{equation}
in the discrete field case, where the sum is finite, and
\begin{equation}
I(\widetilde{\varphi+i\psi)}=X(\varphi)+iX(\psi),
\quad \varphi,\,\psi\in{\cal S} \label{($3.4'$)}
\end{equation}
in the generalized field case.

Obviously,
\begin{eqnarray*}
\left\|\sum c_ne^{i(n,x)}\right\|_{L^2_G}^2
&=&\sum\sum c_n\bar c_m\int e^{i(n-m),x}G(\,dx)\\
&=&\sum\sum c_n\bar c_m EX_nX_m=E\left|\sum c_nX_n\right|^2,
\end{eqnarray*}
and
\begin{eqnarray*}
\|\widetilde{\varphi+i\psi}\|^2_{L^2_G}
&=&\int[\tilde\varphi(x)\bar{\tilde\varphi}(x)
-i\tilde\varphi(x)\bar{\tilde\psi}(x)
+i\tilde\psi(x)\bar{\tilde\varphi}(x)
+\tilde\psi(x)\bar{\tilde\psi}(x)]G(\,dx)\\
&=&EX(\varphi)^2-iEX(\varphi)X(\psi)+iEX(\psi)X(\varphi)
+EX(\psi)^2 \\
&=&E\left(|X(\varphi)+iX(\psi)|\right)^2.
\end{eqnarray*}
This means that the mapping $I$ from a linear subspace of $L_G^2$
to ${\cal H}_1^c$ is norm preserving. Besides, the subspace 
where $I$ was defined is dense in $L^2_G$, since the space of 
continuous functions is dense in $L^2_G$ if $G$ is a finite 
measure on the torus $R^\nu/2\pi\textrm{\BBB Z}_\nu$, and the 
space of continuous functions with a compact support is dense 
in $L^2_G(R^{\nu})$ if the measure~$G$ satisfies 
relation~(\ref{(3.3)}). Hence the mapping $I$ can be uniquely
extended to a norm preserving transformation from $L^2_G$ to
${\cal H}^c_1$. Since the random variables $X_n$ or $X(\varphi)$
are obtained as the image of some element from $L_G^2$ under this
transformation, $I$ is a unitary transformation from $L^2_G$ to
${\cal H}^c_1$. A unitary transformation preserves not only the
norm, but also the scalar product. Hence
$\int f(x)\bar g(x)G(\,dx)=EI(f)\overline{I(g)}$ for all
$f,\,g\in L^2_G$.

Now we define the random spectral measure $Z_G(A)$ for all
$A\in{\cal B}^\nu$ such that $G(A)<\infty$ by the formula
$$
Z_G(A)=I(\chi_A),
$$
where $\chi_A$ denotes the indicator function of the set~$A$. It 
is clear that

\medskip
\begin{description}
\item [(i)] The random variables  $Z_G(A)$ are complex valued, 
jointly Gaussian random variables. (The random variables 
$\textrm{Re}\, Z_G(A)$ and $\textrm{Im}\, Z_G(A)$ with possibly 
different sets~$A$ are jointly Gaussian.)
\item [(ii)] $EZ_G(A)=0$,
\item [(iii)] $EZ_G(A)\overline {Z_G(B)}=G(A\cap B)$,
\item [(iv)] $\sum\limits_{j=1}^nZ_G(A_j)
=Z_G\left(\bigcup\limits_{j=1}^n A_j\right)$ if
$A_1,\dots,A_n$ are disjoint sets.

Also the following relation holds.

\item [(v)] $Z_G(A)=\overline{Z_G(-A)}$.

This follows from the relation
\item [(v$'$)] $I(f)=\overline{I(f_-)}$ for all $f\in L^2_G$, where
$f_-(x)=\overline{f(-x)}$.
\end{description}

Relation (v$'$) can be simply checked if $f$ is a finite
trigonometrical polynomial in the discrete field case, or if
$f=\tilde\varphi$, $\varphi\in{\cal S}^c$,  in the generalized field
case. (In the case $f=\tilde\varphi$, $\varphi\in{\cal S}^c$, the
following argument works. Put
$f(x)=\tilde\varphi_1(x)+i\tilde\varphi_2(x)$ with
$\varphi_1,\varphi_2\in{\cal S}$. Then $I(f)=X(\varphi_1)+iX(\varphi_2)$,
and $f_-(x)=\bar{\tilde\varphi}_1(-x)-i\bar{\tilde\varphi}_2(-x)
=\tilde\varphi_1(x)+i(\widetilde{-\varphi_2}(x)$, hence
$I(f_-)=X(\varphi_1)+iX(-\varphi_2)=X(\varphi_1)-iX(\varphi_2)
=\overline{I(f)}$.)
Then a simple limiting procedure implies~(v$'$) in the general
case. Relation~(iii) follows from the identity
$EZ_G(A)\overline {Z_G(B)}=EI(\chi_A)\overline{I(\chi_B)}
=\int \chi_A(x)\overline{\chi_B(x)}G(\,dx)=G(A\cap B)$. The remaining
properties of $Z_G(\cdot)$ are simple consequences of the definition.

\medskip\noindent
{\it Remark.}\/ Property (iv) could have been omitted from the
definition of random spectral measures, since it follows from
property~(iii). To show this it is enough to check that if
$A_1,\dots,A_n$ are disjoint sets, and property~(iii) holds, then
$$
E\left(\sum_{j=1}^n Z_G(A_j)-Z_G\left(\bigcup\limits_{j=1}^n A_j\right)\right)
\overline{\left(\sum_{j=1}^n Z_G(A_j)-Z_G
\left(\bigcup\limits_{j=1}^n A_j\right)\right)}=0.
$$
\medskip
Now we introduce the following

\medskip\noindent
{\bf Definition of Random Spectral Measure.} 
\index{Gaussian random spectral measure.} 
{\it Let $G$ be a spectral measure. A set of random variables 
$Z_G(A)$, $G(A)<\infty$, satisfying (i)--(v) is called a 
(Gaussian) random spectral measure corresponding to the 
spectral measure~$G$.}

\medskip
Given a Gaussian random spectral measure $Z_G$ corresponding to a
spectral measure $G$ we define the (one-fold) stochastic integral
$\int f(x)Z_G(\,dx)$ for an appropriate class of functions~$f$.
\index{one-fold stochastic integral}
Let us first consider simple functions of the form
$f(x)=\sum c_i\chi_{A_i}(x)$, where the sum is finite, and
$G(A_i)<\infty$ for all indices~$i$. In this case we define
$$
\int f(x)Z_G(\,dx)=\sum c_iZ_G(A_i).
$$
Then we have
\begin{equation}
E\left|\int f(x)Z_G(\,dx)\right|^2=\sum c_i\bar c_jG(A_i\cap A_j)
=\int |f(x)|^2G(\,dx). \label{(3.5)}
\end{equation}

Since the simple functions are dense in $L^2_G$, 
relation~(\ref{(3.5)}) enables us to define 
$\int f(x)Z_G(\,dx)$ for all $f\in L^2_G$ via
$L_2$-continuity. It can be seen that this integral
satisfies the identity
\begin{equation}
E\int f(x)Z_G(\,dx)\overline{\int g(x)Z_G(\,dx)}
=\int f(x)\overline{g(x)}G(\,dx) \label{(3.6a)}  
\end{equation}
for all pairs of functions $f,g\in L_G^2$. Moreover, similar
approximation with simple functions yields that
\begin{equation}
\int f(x)Z_G(\,dx)=\overline{\int\overline{f(-x)}Z_G(\,dx)}
\label{(3.6b)}
\end{equation}
for a function $f\in L^2_G$. Here we exploit the identity 
$Z_G(A)=\overline{Z_G(-A)}$ formulated in property~(v) of 
the random spectral measure $Z_G$.

The last two identities together with the relations~(\ref{(3.1)}) 
and~(\ref{(3.2)}) imply that if we define the set of random 
variables $X_n$ and $X(\varphi)$ by means of the formula 
\begin{equation}
X_n=\int e^{i(n,x)}Z_G(\,dx), 
\quad n\in\textrm{\BBB Z}_\nu, 
\label{(3.6)}
\end{equation}
and
\begin{equation}
X(\varphi)=\int\tilde\varphi(x) Z_G(\,dx), 
\quad \varphi\in{\cal S},
\label{($3.6'$)}
\end{equation}
where we integrate with respect to the random spectral 
measure $Z_G$, then we get a Gaussian stationary 
random discrete and generalized field with spectral 
measure~$G$, i.e. with correlation function given in 
formulas~(\ref{(3.1)}) and~(\ref{(3.2)}). To check this 
statement first we have to show that the random 
variables $X_n$ and $X(\varphi)$ defined in (\ref{(3.6)}) 
and~(\ref{($3.6'$)}) are real valued, or equivalently 
saying the identities $X_n=\overline{X_n}$ and 
$X(\varphi)=\overline{X(\varphi)}$ hold with 
probability~1. This follows from relation~(\ref{(3.6b)}) 
and the identities $e^{i(n,x)}=\overline{e^{(i(n,-x)}}$ 
and $\tilde\varphi(x)=\overline{\tilde\varphi(-x)}$ 
for a (real valued) function $\varphi\in{\cal S}$. 
Then we can calculate the correlation functions 
$EX_nX_m=EX_n\overline{X}_m$ and 
$EX(\varphi)X(\psi)=EX(\varphi)\overline{X(\psi)}$
by means of formula~(\ref{(3.6a)}), (\ref{(3.6)})
and~(\ref{($3.6'$)}).

We also have
$$
\int f(x)Z_G(\,dx)=I(f) \quad \textrm{for all } f\in L_G^2
$$
if we consider the previously defined mapping $I(f)$ with the
stationary random fields defined in~(\ref{(3.6)}) 
and~(\ref{($3.6'$)}). Now we
formulate the following

\medskip\noindent
{\bf Theorem 3.1.} {\it For a stationary Gaussian random field (a
discrete or generalized one) with a spectral measure $G$ there exists
a unique Gaussian random spectral measure $Z_G$ corresponding to the
spectral measure~$G$ on the same probability space as the Gaussian
random field such that relation~(\ref{(3.6)}) 
or~(\ref{($3.6'$)}) holds in the
discrete or generalized field case respectively.

Furthermore
\begin{equation}
{\cal B}(Z_G(A),\; G(A)<\infty)=\left\{
\begin{array}{l}
{\cal B}(X_n,\;n\in\textrm{\BBB Z}_\nu) 
\textrm{ in the discrete field case,}\\
{\cal B}(X(\varphi),\;\varphi\in{\cal S})
\textrm{ in the generalized field case.}
\end{array} \right. \label{(3.7)}
\end{equation}
}

\medskip
If a stationary Gaussian random field $X_n$, 
$n\in\textrm{\BBB Z}_\nu$, or $X(\varphi)$, 
$\varphi\in{\cal S}$, and a random spectral measure $Z_G$ 
satisfy relation~(\ref{(3.6)}) or~(\ref{($3.6'$)}), then 
we say that this random spectral measure is adapted to
this Gaussian random field.
\index{random spectral measure adapted to a Gaussian 
random field}

\medskip\noindent
{\it Proof of Theorem 3.1.}\/ Given a stationary Gaussian 
random field (discrete or stationary one) with a spectral 
measure~$G$, we have constructed a random spectral measure 
$Z_G$ corresponding to the spectral measure~$G$. Moreover, 
the random integrals given in formulas~(\ref{(3.6)}) 
or~(\ref{($3.6'$)}) define the original stationary random
field. Since all random variables $Z_G(A)$ are measurable with
respect to the original random field, relation~(\ref{(3.6)}) 
or~(\ref{($3.6'$)}) implies~(\ref{(3.7)}).

To prove the uniqueness, it is enough to observe that because 
of the linearity and $L_2$ continuity of stochastic integrals 
relation~(\ref{(3.6)}) or~(\ref{($3.6'$)}) implies that
$$
Z_G(A)=\int \chi_A(x)Z_G(\,dx)=I(\chi_A)
$$
for a Gaussian random spectral measure corresponding to the spectral
measure~$G$ appearing in Theorem~3.1. \hfill$\qed$

\medskip
Finally we list some additional properties of Gaussian 
random spectral measures.
\index{Gaussian random spectral measure.} 

\medskip
\begin{description}
\item[(vi)] The random variables $\textrm{Re}\, Z_G(A)$ are 
independent of the random variables $\textrm{Im}\, Z_G(A)$.
\item[(vii)] The random variables of the form $Z_G(A\cup(-A))$ 
are real valued. If the sets $A_1\cup(-A_1)$,\dots, 
$A_n\cup(-A_n)$ are disjoint, then the random variables 
$Z_G(A_1)$,\dots, $Z_G(A_n)$ are independent.
\item[(viii)] The relations 
$\textrm{Re}\, Z_G(-A)=\textrm{Re}\, Z_G(A)$ and 
$\textrm{Im}\, Z_G(-A)=-\textrm{Im}\, Z_G(A)$ hold, and if 
$A\cap(-A)=\emptyset$, then the (Gaussian) random variables 
$\textrm{Re}\, Z_G(A)$ and $\textrm{Im}\, Z_G(A)$ are 
independent with expectation zero and variance $G(A)/2$.
\end{description}

\medskip
These properties easily follow from (i)--(v). Since $Z_G(\cdot)$
are complex valued Gaussian random variables, to prove the above
formulated independence it is enough to show that the real and 
imaginary parts are uncorrelated. We show, as an example, 
the proof of~(vi).
\begin{eqnarray*}
E\textrm{Re}\, Z_G(A)\textrm{Im}\, Z_G(B)&&=\frac1{4i} 
E(Z_G(A)+\overline{Z_G(A)})
(Z_G(B)-\overline{Z_G(B)})\\
&&=\frac1{4i}E(Z_G(A)+Z_G(-A))(\overline{Z_G(-B)}-\overline{Z_G(B)})\\
&&=\frac1{4i}G(A\cap(-B))-\frac1{4i}G(A\cap B)\\
&&\qquad+\frac1{4i}G((-A)\cap(-B))-\frac1{4i}G((-A)\cap B)=0
\end{eqnarray*}
for all pairs of sets $A$ and $B$ such that $G(A)<\infty$, $G(B)<\infty$,
since $G(D)=G(-D)$ for all $D\in{\cal B}^\nu$. The fact that
$Z_G(A\cup(-A))$ is real valued random variable, and the relations
$\textrm{Re}\, Z_G(-A)=\textrm{Re}\, Z_G(A)$, 
$\textrm{Im}\, Z_G(-A)=-\textrm{Im}\, Z_G(A)$ under the conditions
of~(viii) follow directly from~(v). The remaining statements of~(vii)
and~(viii) can be proved similarly to~(vi) only the calculations are
simpler in this case.

The properties of the random spectral measure $Z_G$ listed above imply
in particular that the spectral measure~$G$ determines the joint
distribution of the corresponding random variables $Z_G(B)$,
$B\in{\cal B}^\nu$.

\section{On the Spectral Representation of the Covariance 
Function of Stationary Random Fields}

The results formulated under the name of Bochner and 
Bochner--Schwartz theorem (I write this, because actually 
I presented not these theorems but an important consequence 
of them) have the following content. Given a finite, even 
measure~$G$ on the torus
$R^{\nu}/2\pi\textrm{\BBB Z}_\nu$ one can define a (Gaussian)
discrete stationary field with correlation function 
satisfying~(\ref{(3.1)})
with this measure~$G$. For an even measure $G$ on~$R^\nu$
satisfying~(\ref{(3.3)}) there exists a (Gaussian) generalized 
stationary field with correlation function defined in 
formula~(\ref{(3.2)}) with this measure $G$. The Bochner and 
Bochner--Schwartz theorems state that the correlation function 
of all (Gaussian) discrete stationary fields, respectively of 
all stationary generalized fields can be represented
in such a way. Let us explain this in more detail.

First I formulate the following

\medskip\noindent
{\bf Proposition~3C.} {\it Let $G$ be a finite measure 
on the torus $R^{\nu}/2\pi\textrm{\BBB Z}_\nu$ such that 
$G(A)=G(-A)$ for all measurable sets~$A$. Then there 
exists a Gaussian discrete stationary random field 
$X_n$, $n\in\textrm{\BBB Z}_\nu$, with expectation zero 
such that its correlation function $r(n)=EX_kX_{k+n}$, 
$n,k\in\textrm{\BBB Z}_\nu$, is given by
formula~(\ref{(3.1)}) with this measure~$G$.

Let $G$ be a measure on~$R^\nu$ satisfying~(\ref{(3.3)}) 
and such that $G(A)=G(-A)$ for all measurable sets~$A$. 
Then there exists a Gaussian stationary generalized 
random field $X(\varphi)$, $\varphi\in{\cal S}$, with
expectation $EX(\varphi)=0$ for all $\varphi\in{\cal S}$ 
such that its covariance function $EX(\varphi)X(\psi)$, 
$\varphi,\psi\in{\cal S}$, satisfies formula~(\ref{(3.2)}) 
with this measure~$G$.

Moreover, the correlation function $r(n)$ or $EX(\varphi)X(\psi)$,
$\varphi,\psi\in{\cal S}$, determines the measure~$G$ uniquely.}

\medskip\noindent
{\it Proof of Proposition~3C.}\/ By Kolmogorov's theorem about
the existence of random processes with consistent finite 
dimensional distributions it is enough to prove the following 
statement to show the existence of the Gaussian discrete 
stationary field with the demanded properties. For any points
$n_1,\dots,n_p\in\textrm{\BBB Z}_\nu$ there exists a Gaussian 
random vector $(X_{n_1},\dots,X_{n_p})$ with expectation zero 
and covariance matrix $EX_{n_j}X_{n_k}=r(n_j-n_k)$. (Observe 
that the function~$r(n)$ is real valued, $r(n)=r(-n)$, because 
of the evenness of the spectral measure~$G$.) Hence it is enough 
to check that the corresponding matrix is positive definite, 
i.e. $\sum\limits_{j,k} c_jc_k r(n_j-n_k)\ge0$ for all real 
vectors $(c_1,\dots,c_p)$. This relation holds, because
$\sum\limits_{j,k} c_jc_k r(n_j-n_k)=\int
|\sum\limits_j c_je^{i(n_j,x)}|^2\,G(\,dx)\ge0$ by 
formula~(\ref{(3.1)}).

It can be proved similarly that in the generalized field 
case there exists a Gaussian random field with expectation 
zero whose covariance function satisfies 
formula~(\ref{(3.2)}). (Let us observe that the relation 
$G(A)=G(-A)$ implies that $EX(\varphi)X(\psi)$ is a real 
number for all $\varphi,\,\psi\in{\cal S}$, since
$EX(\varphi)X(\psi)=\overline{EX(\varphi)X(\psi)}$ in 
this case. In the proof of this identity we exploit that
$\bar{\tilde f}(x)=\tilde f(-x)$ for a real valued 
function~$f$.) We also have to show that a random field 
with such a distribution is a generalized field, i.e. it 
satisfies properties~(a) and~(b) given in the definition 
of generalized fields. 

It is not difficult to show that if $\varphi_n\to\varphi$ 
in the topology of the space ${\cal S}$, then 
$E[X(\varphi_n)-X(\varphi)]^2
=\int|\tilde\varphi_n(x)-\tilde\varphi(x)|^2 G(\,dx)\to0$ 
as $n\to\infty$, hence property~(b) holds. (Here we exploit 
that the transformation $\varphi\to\tilde\varphi$ is 
bicontinuous in the space~${\cal S}$.) Property~(a) also 
holds, because, as it is not difficult to check with the 
help of formula~(\ref{(3.2)}),
\begin{eqnarray*}
&&E[a_1X(\varphi_1)+a_2X(\varphi_2)
-X(\varphi(a_1\varphi_1+a_2\varphi_2)]^2\\
&&\qquad =\int\left|a_1\tilde\varphi_1(x)+a_2\tilde\varphi_2(x)
-(\widetilde{a_1\varphi_1+a_2\varphi_2})(x)\right|^2G(\,dx)=0.
\end{eqnarray*}

It is clear that the Gaussian random field constructed in 
such a way is stationary.

Finally, as we have seen in our considerations in the main 
text, the correlation function determines the integral 
$\int f(x)\,G(\,dx)$ for all continuous functions~$f$ with 
a bounded support, hence it also determines the 
measure~$G$. \hfill$\qed$

\medskip
The Bochner and Bochner--Schwartz theorems enable us to 
show that the correlation function of all stationary 
(Gaussian) random fields (discrete or generalized one) 
can be represented in the above way with an appropriate 
spectral measure~$G$. To see this let us formulate these 
results in their original form.

To formulate Bochner's theorem first we introduce the 
following notion.

\medskip\noindent
{\bf Definition of Positive Definite Functions.} 
\index{positive  definite function}
{\it Let $f(x)$ be a (complex valued) function on 
$\textrm{\BBB Z}_\nu$ (or on $R^\nu$). We say that  $f(\cdot)$ 
is a positive definite function if for all parameters~$p$, 
complex numbers $c_1,\dots,c_p$ and points $x_1,\dots,x_p$ in  
$\textrm{\BBB Z}_\nu$ (or in $R^\nu$) the inequality
$$
\sum_{j=1}^p\sum_{k=1}^p c_j\bar c_k f(x_j-x_k)\ge0
$$
holds.}

\medskip
A simple example for positive definite functions is the function
$f(x)=e^{i(t,x)}$, where $t\in\textrm{\BBB Z}_\nu$ in the discrete, 
and $t\in R^\nu$ in the continuous case. Bochner's theorem provides 
a complete description of positive definite functions.

\medskip\noindent
{\bf Bochner's Theorem. (Its Original Form.)} 
\index{Bochner theorem}
{\it A complex valued function $f(x)$ defined on 
$\textrm{\BBB Z}_\nu$ is positive definite if and only if it can 
be written in the form $f(x)=\int e^{i(t,x)}G(\,dx)$ for all
$x\in\textrm{\BBB Z}_\nu$ with a finite measure~$G$ on the
torus~$R^\nu/2\pi\textrm{\BBB Z}_\nu$. The measure~$G$ is uniquely
determined.

A complex valued function $f(x)$ defined on $R^\nu$ is positive 
definite and continuous at the origin  if and only if it 
can be written in the form $f(x)=\int e^{i(t,x)}G(\,dx)$ 
for all $x\in R^\nu$ with a finite measure~$G$ on $R^\nu$. 
The measure~$G$ is uniquely determined.}

\medskip
It is not difficult to see that the covariance function 
$r(n)=EX_kX_{k+n}$, ($EX_n=0$), $k,n\in\textrm{\BBB Z}_\nu$, 
of a stationary (Gaussian) random field~$X_n$ is a positive 
definite function, since
$\sum\limits_{j,k} c_j\bar c_kr(n_j-n_k)=E|\sum\limits_j c_jX_{n_j}|^2>0$
for any vector $(c_1,\dots,c_p)$. Hence Bochner's theorem can be applied
for it. Besides, the relation $r(n)=r(-n)$ together with the
uniqueness of the measure~$G$ appearing in Bochner's theorem imply that
the identity $G(A)=G(-A)$ holds for all measurable sets~$G$. This
implies the result formulated in the main text under the name Bochner's
theorem.

\medskip
The Bochner--Schwartz theorem yields an analogous representation
of positive definite generalized functions in~${\cal S}'$ as the
Fourier transforms of positive generalized functions in~${\cal S}'$.
It also states a similar result about generalized functions in the
space~${\cal D}'$. To formulate it we have to introduce some 
definitions. First we have to clarify what a positive generalized 
function is. We introduce this notion both in the 
space~${\cal S}'$ and ${\cal D}'$, and then we characterize them 
in a Theorem.

\medskip\noindent
{\bf Definition of Positive Generalized Functions.}
\index{positive generalized function} 
{\it A continuous linear functional $F\in{\cal S}'$ (or $F\in{\cal D}'$) is 
called a positive generalized function if for all such 
$\varphi\in{\cal S}$ (or $\varphi\in{\cal D}$) test functions 
for which $\varphi(x)\ge0$ for all $x\in R^\nu$ $(F,\varphi)\ge0$.}

\medskip\noindent
{\bf Theorem About the Representation of Positive Generalized
Functions.} {\it All positive generalized functions 
$F\in{\cal S}'$ can be given in the form 
$(F,\varphi)=\int \varphi(x)\mu(\,dx)$, where $\mu$ is a 
polynomially increasing measure on $R^\nu$, i.e.\ it satisfies 
the relation $\int(1+|x|^2)^{-p}\mu(\,dx)<\infty$ with
some $p>0$. Similarly, all positive generalized functions in
${\cal D}'$ can be given in the form
$(F,\varphi)=\int \varphi(x)\mu(\,dx)$ with such a measure 
$\mu$ on $R^\nu$ which is finite in all bounded regions. The 
positive generalized function~$F$ uniquely determines the 
measure~$\mu$ in both cases.}

\medskip
We also introduce a rather technical notion and formulate a 
result about it. Let us remark that if $\varphi\in{\cal S}^c$ 
and $\psi\in{\cal S}^c$, then also their product 
$\varphi\psi\in{\cal S}^c$. In particular, 
$\varphi\bar\varphi=|\varphi|^2\in{\cal S}$ if 
$\varphi\in{\cal S}^c$. The analogous result also holds in 
the space ${\cal D}$.

\medskip\noindent
{\bf Definition of Multiplicatively Positive Generalized Functions.}
\index{multiplicatively positive generalized function}
{\it A generalized function ${\cal F}\in{\cal S}'$ 
(or $F\in{\cal D}'$) is multiplicatively positive if
$(F,\varphi\bar\varphi)=(F,|\varphi|^2)\ge0$ for all
$\varphi\in{\cal S}^c$ (or in $\varphi\in{\cal D}$).}

\medskip\noindent
{\bf Theorem About the Characterization of Multiplicatively Positive
Generalized Functions.} {\it A generalized function $F\in{\cal S}'$
(or $F\in{\cal D}'$) is multiplicatively positive if and only if it is
positive.}

\medskip
Now I introduce the definition of positive definite generalized
functions.

\medskip\noindent
{\bf Definition of Positive Definite Generalized Functions.} 
\index{positive definite generalized function} 
{A generalized function $F\in{\cal S}'$ (or $F\in{\cal D}'$) is 
positive definite if $(F,\varphi*\varphi^*)\ge0$ for all 
$\varphi\in{\cal S}^c$ (of $\varphi\in{\cal D}$), where 
$\varphi^*(x)=\overline {\varphi(-x)}$, and $*$ denotes 
convolution, i.e. $\varphi*\varphi^*(x)=\int\varphi(t)
\overline{\varphi(t-x)}\,dt$.}

\medskip
We refer to~\cite{r15} for an explanation why this 
definition of positive definite generalized functions is 
natural. Let us remark that if $\varphi,\psi\in{\cal S}^c$, 
then $\varphi*\psi\in{\cal S}^c$, and the analogous result 
holds in~${\cal D}$. The original version of the 
Bochner--Schwartz theorem has the following form.

\medskip\noindent
{\bf Bochner--Schwartz Theorem. (Its Original Form.)} 
\index{Bochner--Schwartz theorem} 
{\it Let $F$ be a positive definite generalized function in 
the space ${\cal S}'$ (or ${\cal D}'$). Then it is the Fourier 
transform of a polynomially increasing measure~$\mu$ on 
$R^\nu$, i.e. the identity 
$(F,\varphi)=\int\tilde\varphi(x)\,\mu(\,dx)$ holds for all
$\varphi\in{\cal S}^c$ (or $\varphi\in{\cal D}$) with a 
measure $\mu$ that satisfies the relation 
$\int(1+|x|^2)^{-p}\mu(\,dx)<\infty$ with an appropriate 
$p>0$. The generalized function~$F$ uniquely determines the 
measure~$\mu$. On the other hand, if $\mu$ is a polynomially 
increasing measure on $R^\nu$, then the formula
$(F,\varphi)=\int \tilde\varphi(x)\mu(\,dx)$ with 
$\varphi\in{\cal S}^c$ (or $\varphi\in{\cal D}$) defines 
a positive definite generalized function~$F$ in the space 
${\cal S}'$ (or ${\cal D}'$).}

\medskip\noindent
{\it Remark.} It is a remarkable and surprising fact that the class
of positive definite generalized functions are represented by the
same class of measures~$\mu$ in the spaces ${\cal S}'$ and ${\cal D}'$.
(In the representation of positive generalized functions the class
of measures~$\mu$ considered in the case of ${\cal D}'$ is much
larger, than in the case of ${\cal S}'$.) Let us remark that in the
representation of the positive definite generalized functions in
${\cal D}'$ the function $\tilde\varphi$ we integrate is not in the
class~${\cal D}$, but in the space~${\cal Z}$ consisting of the Fourier
transforms of the functions in~${\cal D}$.

\medskip
It is relatively simple to prove the representation of positive
definite generalized functions given in the Bochner--Schwartz
theorem for the class~${\cal S}'$. Some calculation shows that if $F$ is
a positive definite generalized function, then its Fourier transform
is a multiplicatively positive generalized function. Indeed, since
the Fourier transform of the convolution $\varphi*\psi(x)$ equals
$\tilde\varphi(t)\tilde\psi(t)$, and the Fourier transform of
$\varphi^*(x)=\overline{\varphi(-x)}$ equals $\overline{\tilde\varphi(t)}$,
the Fourier transform of $\varphi*\varphi^*(x)$ equals
$\tilde\varphi(t)\bar{\tilde\varphi}(t)$. Hence the positive
definitiveness property of the generalized function~$F$ and the
definition of the Fourier transform of generalized functions imply that
$(\tilde F,\tilde\varphi\bar{\tilde\varphi})
=(2\pi)^{\nu}(F,\varphi*\varphi^*)\ge0$ for all $\varphi\in{\cal S}^c$.
Since every function of~${\cal S}^c$ is the Fourier transform
$\tilde\varphi$ of some function $\varphi\in{\cal S}^c$ this implies
that $\tilde F$ is a multiplicatively positive and as a consequence
a positive generalized function in ${\cal S}'$. Such generalized
functions have a good representation with the help of a polynomially
increasing positive measure~$\mu$. Since
$(F,\varphi)=(2\pi)^{-\nu}(\tilde F,\tilde\varphi)$ it is not difficult
to prove the Bochner--Schwartz theorem for the space~${\cal S}'$ with the
help of this fact. The proof is much harder if the space~${\cal D}'$
is considered, but we do not need that result.

The Bochner--Schwartz theorem in itself is not sufficient 
to describe the correlation function of a generalized 
random field. We still need another important result of 
Laurent Schwartz which gives useful information about 
the behaviour of (Hermitian) bilinear functionals in 
${\cal S}^c$ and some additional information about the 
behaviour of translation invariant (Hermitian) bilinear 
functionals in this space. To formulate these results 
first we introduce the following definition.

\medskip\noindent
{\bf Definition of Hermitian Bilinear and Translation Invariant
Hermitian Bilinear Functionals in the Space ${\cal S}^c$.} {\it 
A function $B(\varphi,\psi)$, $\varphi,\psi\in{\cal S}^c$, is a 
Hermitian bilinear functional in the space ${\cal S}^c$ if for 
all fixed $\psi\in{\cal S}^c$ $B(\varphi,\psi)$ is a continuous 
linear functional of the variable~$\varphi$ in the topology 
of~${\cal S}^c$, and for all fixed $\varphi\in{\cal S}^c$ 
$\overline{B(\varphi,\psi)}$ is a continuous linear functional 
of the variable~$\psi$ in the topology of~${\cal S}^c$.

A Hermitian bilinear functional $B(\varphi,\psi)$ in 
${\cal S}^c$ is translation invariant if it does not 
change by a simultaneous shift of its variables $\varphi$ 
and $\psi$, i.e.\ if 
$B(\varphi(x),\psi(x))=B(\varphi(x-h),\psi(x-h))$ for 
all $h\in R^\nu$.}

\medskip\noindent
{\bf Definition of Positive Definite Hermitian Bilinear 
Functionals.}
\index{positive definite Hermitian bilinear functional}
{\it We say that a Hermitian bilinear functional 
$B(\varphi,\psi)$ in ${\cal S}^c$ is positive definite if 
$B(\varphi,\varphi)\ge0$ for all $\varphi\in{\cal S}^c$.}

\medskip
The next result characterizes the Hermitian bilinear and 
translation invariant Hermitian bilinear functionals 
in~${\cal S}^c$.

\medskip\noindent
{\bf Theorem 3D.} {\it All Hermitian bilinear functionals
$B(\varphi,\psi)$ in ${\cal S}^c$ can be given in the form
$B(\varphi,\psi)=(F_1,\varphi(x)\overline{\psi(y)})$,
$\varphi,\psi\in{\cal S}^c$, where $F_1$ is a continuous 
linear functional on ${\cal S}^c\times{\cal S}^c$, i.e. it 
is a generalized function in~${{\cal S}_{2\nu}}'$.

A translation invariant Hermitian bilinear functional 
in ${\cal S}^c$ can be given in the form 
${\cal B}(\varphi,\psi)=(F,\varphi*\psi^*)$,
$\varphi,\psi\in{\cal S}^c$, where $F\in{\cal S}$,
$\psi^*(x)=\overline\psi(-x)$, and $*$ denotes convolution.

The Hermitian bilinear form $B(\varphi,\psi)$ determines 
the generalized functions $F_1$ uniquely, and if it is 
translation invariant, then the same can be told about 
the generalized function $F$. Besides, for  all functionals
$F_1\in{\cal S}_{2\nu}'$ and $F\in{\cal S}'$ the above 
formulas define a Hermitian bilinear functional and a 
translation invariant Hermitian bilinear functional 
in~${\cal S}_\nu^c$ respectively.}

\medskip
Let us consider a Gaussian generalized random field 
$X(\varphi)$, $\varphi\in{\cal S}$, with expectation 
zero together with its correlation function 
$B(\varphi,\psi)=EX\varphi)X(\psi)$, \
$\varphi,\psi\in{\cal S}$. More precisely, let us 
consider the complexification 
$X(\varphi_1+i\varphi_2)=X(\varphi_1)+iX(\varphi_2)$
of this random field and its correlation function
$B(\varphi,\psi)=EX(\varphi)\overline{X(\psi)}$,
$\varphi,\psi\in{\cal S}^c$. This correlation function
$B(\varphi,\psi)$ is a translation invariant Hermitian 
bilinear functional in ${\cal S}^c$, hence it can be 
written in the form $B(\varphi,\psi)=(F,\varphi*\psi^*)$ 
with an appropriate $F\in{\cal S}'$. Moreover, 
$B(\varphi,\varphi)\ge0$ for all $\varphi\in{\cal S}^c$, 
and this means that the generalized function 
$F\in{\cal S}'$ corresponding to $B(\varphi,\psi)$ is
positive definite. Hence the Bochner--Schwartz theorem 
can be applied for it, and it yields that
$$
EX(\varphi)X(\psi)=\int \widetilde{\varphi*\psi^*}(x)\,G(\,dx)
=\int \tilde\varphi(x)\bar{\tilde\psi}(x)\,G(\,dx) 
\quad\textrm{for all } \varphi,\,\psi\in {\cal S}^c
$$
with a uniquely determined, polynomially increasing 
measure~$G$ on~$R^\nu$. Now we complete the proof 
of Theorem~3B with the help of these results.

\medskip\noindent
{\it Proof of Theorem 3B.}\/ We have already proved 
relations~(\ref{(3.2)}) and~(\ref{(3.3)}) with the 
help of some  results about generalized functions. 
To complete the proof of  Theorem~3B we still  have 
to show that $G$ is an even measure.  In the proof 
of this statement we exploit that for a real valued 
function $\varphi\in{\cal S}$ the random variable 
$X(\varphi)$  is also real valued. Hence if 
$\varphi,\psi\in{\cal S}$, then 
$EX(\varphi)X(\psi)=\overline{EX(\varphi)X(\psi)}$. 
Besides, $\tilde\varphi(-x)=\bar{\tilde\varphi}(x)$ 
and $\tilde\psi(-x)=\bar{\tilde\psi}(x)$ in this case. 
Hence
\begin{eqnarray*}
\int \tilde\varphi(x)\bar{\tilde\psi}(x)\,G(\,dx)
&=&\int \bar{\tilde\varphi}(x)\tilde\psi(x)\,G(\,dx)\\
&=&\int \tilde\varphi(-x)\bar{\tilde\psi}(-x)\,G(\,dx)
=\int \tilde\varphi(x)\bar{\tilde\psi}(x)\,G^-(\,dx)
\end{eqnarray*}
for all $\varphi,\psi\in{\cal S}$, where $G^-(A)=G(-A)$ 
for all $A\in{\cal B}^\nu$. This relation implies that 
the measures~$G$ and~$G^-$ agree. The proof of 
Theorem~3B is completed. \hfill$\qed$

\chapter{Multiple Wiener--It\^o Integrals}

In this chapter we define the so-called multiple Wiener--It\^o
integrals, and we prove their most important properties with the
help of It\^o's formula, whose proof is postponed to the next
chapter. More precisely, we discuss in this chapter a modified
version of the Wiener--It\^o integrals with respect to a random
spectral measure rather than with respect to a random measure with
independent increments. This modification makes it necessary to
slightly change the definition of the integral. This modified
Wiener--It\^o integral seems to be a more useful tool than the
original one or the Wick polynomials in the study of the 
problems in this work, because it enables us to describe the 
action of shift transformations.

Let $G$ be the spectral measure of a stationary Gaussian field
(discrete or generalized one). We define the following
{\it real}\/ Hilbert spaces $\bar{{\cal H}}_G^n$ and ${\cal H}_G^n$,
$n=1,2,\dots$. We have $f_n\in\bar{{\cal H}}_G^n$ if and only if
$f_n=f_n(x_1,\dots,x_n)$, \ $x_j\in R^\nu$, $j=1,2,\dots,n$, is a
complex valued function of $n$ variables, and

\medskip
\begin{description}
\item[(a)] $f_n(-x_1,\dots,-x_n)=\overline{f_n(x_1,\dots,x_n)}$,
\item[(b)]
$\|f_n\|^2=\int|f_n(x_1,\dots,x_n)|^2G(\,dx_1)\dots G(\,dx_n)<\infty$.
\end{description}

\medskip
Relation~(b) also defines the norm in $\bar{{\cal H}}^n_G$. The
subspace ${\cal H}^n_G\subset\bar{{\cal H}}_G^n$ contains those functions
$f_n\in\bar{{\cal H}}_G^n$ which are invariant under permutations of
their arguments, i.e.

\medskip
\begin{description}
\item[(c)] $f_n(x_{\pi(1)},\dots,x_{\pi(n)}))=f_n(x_1,\dots,x_n)$
for all $\pi\in\Pi_n$, where $\Pi_n$ denotes the group of all
permutations of the set $\{1,2,\dots,n\}$.
\end{description}

\medskip
The norm in ${\cal H}_G^n$ is defined in the same way as in
$\bar{{\cal H}}_G^n$. Moreover, the scalar product is also similarly
defined, namely if $f,\,g\in\bar{{\cal H}}_G^n$, then
\begin{eqnarray*}
(f,g)&=&\int f(x_1,\dots,x_n)\overline{g(x_1,\dots,x_n)}
G(\,dx_1)\dots G(\,dx_n)\\
&=&\int f(x_1,\dots,x_n)g(-x_1,\dots,-x_n)G(\,dx_1)\dots G(\,dx_n).
\end{eqnarray*}
Because of the symmetry $G(A)=G(-A)$ of the spectral measure
$(f,g)=\overline{(f,g)}$, i.e. the scalar product $(f,g)$ is a real
number for all $f,\,g\in\bar{{\cal H}}_G^n$. This means that
$\bar{{\cal H}}_G^n$ is a real Hilbert space.
We also define ${\cal H}_G^0=\bar{{\cal H}}_G^0$ as
the space of real constants with the norm $\|c\|=|c|$.  We remark
that $\bar{{\cal H}}_G^n$ is actually the $n$-fold direct product of
${\cal H}_G^1$, while ${\cal H}_G^n$ is the $n$-fold symmetrical direct
product of ${\cal H}^1_G$. Condition~(a) means heuristically that
$f_n$ is the Fourier transform of a real valued function.

Finally we define the so-called Fock space $\textrm{Exp\,}{\cal H}_G$
whose elements are sequences of functions $f=(f_0,f_1,\dots)$, 
$f_n\in{\cal H}_G^n$ for all $n=0,1,2,\dots$, such that
$$
\|f\|^2=\sum_{n=0}^\infty \frac1{n!}\|f_n\|^2<\infty.
$$
Given a function $f\in\bar{{\cal H}}^n_G$ we define 
$\textrm{Sym}\, f$ as
$$
\textrm{Sym}\, f(x_1,\dots,x_n)=\frac1{n!}\sum_{\pi\in\Pi_n}
f(x_{\pi(1)},\dots,x_{\pi(n)}).
$$
Clearly, $\textrm{Sym}\, f\in{\cal H}_G^n$, and
\begin{equation}
\|\textrm{Sym}\, f\|\le \|f\|. \label{(4.1)}
\end{equation}
\index{Fock space}

Let $Z_G$ be a Gaussian random spectral measure corresponding 
to the spectral measure~$G$ on a probability space 
$(\Omega,{\cal A},P)$. We shall define the $n$-fold 
Wiener--It\^o integrals
$$
I_G(f_n)=\frac1{n!}\int f_n(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)
\quad \textrm{for } f_n\in\bar{{\cal H}}_G^n
$$
and
$$
I_G(f)=\sum_{n=0}^\infty I_G(f_n)\quad \textrm{for }
f=(f_0,f_1,\dots)\in\textrm{Exp}\,{\cal H}_G.
$$
We shall see that  $I_G(f_n)=I_G(\textrm{Sym}\, f_n)$ for all
$f_n\in\bar{{\cal H}}_G^n$. Therefore, it would have been 
sufficient to define the Wiener--It\^o integral only for 
functions in ${\cal H}_G^n$. Nevertheless, some arguments 
become simpler if we work in $\bar{{\cal H}}_G^n$. In the 
definition of Wiener--It\^o integrals first we restrict 
ourselves to the case when the spectral measure is
non-atomic, i.e. $G(\{x\})=0$ for all $x\in R^\nu$. This 
condition is satisfied in all interesting cases. However, 
we shall later show how one can get rid of this restriction.

First we introduce the notion of regular systems for some
collections of subsets of $R^\nu$, define a subclass
$\hat{\bar{{\cal H}}}_G^n\subset\bar{{\cal H}}_G^n$ of simple 
functions with their help, and define the Wiener--It\^o 
integrals for the functions of this subclass.

\medskip\noindent
{\bf Definition of Regular Systems and the Class of Simple 
Functions.} {\it Let 
$${\cal D}=\{\Delta_j,\;j=\pm1,\pm2,\dots,\pm N\}
$$ 
be a finite collection of bounded, measurable sets in $R^\nu$  
indexed by the integers $\pm1,\dots,\pm N$. We say that 
${\cal D}$ is a regular system if $\Delta_j=-\Delta_{-j}$, and 
$\Delta_j\cap\Delta_l=\emptyset$ if $j\neq l$ for all 
$j,l=\pm1,\pm2,\dots,\pm N$. 
\index{regular system of sets in $R^\nu$} 
A function $f\in\bar{{\cal H}}_G^n$ is adapted to this system 
${\cal D}$ if $f(x_1,\dots,x_n)$ is constant on the sets
$\Delta_{j_1}\times\Delta_{j_2}\times\cdots\times\Delta_{j_n}$, \
$j_l=\pm1,\dots,\pm N$, $l=1,2,\dots,n$,  it vanishes outside 
these sets and also on those sets of the form
$\Delta_{j_1}\times\Delta_{j_2}\times\cdots\times\Delta_{j_n}$, 
for which $j_l=\pm j_{l'}$ for some $l\neq l'$. 

A function $f\in\bar{{\cal H}}_G^n$ is in the class
$\hat{\bar{{\cal H}}}_G^n$ of simple functions, 
and a (symmetric) function $f\in{\cal H}_G^n$ is in 
the class $\hat{{\cal H}}_G^n$ of simple symmetric 
functions if it is adapted to some regular system 
${\cal D}=\{\Delta_j,\;j=\pm1,\dots,\pm N\}$.} 
\index{simple function}

\medskip\noindent
{\bf Definition of Wiener--It\^o Integral of Simple Functions.}
{\it 
Let a simple function $f\in\hat{\bar{{\cal H}}}_G^n$ be adapted to
some regular systems ${\cal D}=\{\Delta_j,\;j\pm1,\dots,\pm N\}$. 
Its Wiener--It\^o integral with respect to the random spectral
measure $Z_G$ is defined as
\begin{eqnarray}
&&\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n) 
\label{(4.2)} \\
&&\qquad =n!I_G(f)=\sum_{\substack{j_l=\pm1,\dots,\pm N\\ l=1,2,\dots,n}}
f(x_{j_1},\dots,x_{j_n})Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\nonumber,
\end{eqnarray}
where $x_{j_l}\in\Delta_{j_l}$, $j_l=\pm1,\dots,\pm N$, $l=1,\dots,n$.}
\medskip

We remark that although the regular system ${\cal D}$ to which 
$f$ is adapted, is not uniquely determined (the elements of 
${\cal D}$ can be divided to smaller sets), the integral 
defined in~(\ref{(4.2)}) is meaningful, i.e. it does not 
depend on the choice of ${\cal D}$. This can be seen by
observing that a refinement of a regular system ${\cal D}$ 
to which the function $f$ is adapted yields the same value 
for the sum defining $n!I_G(f)$ in formula~(\ref{(4.2)}) as 
the original one. This follows from the additivity of the 
random spectral measure $Z_G$ formulated in its 
property~(iv), since this 
implies that each term
$f(x_{j_1},\dots,x_{j_n})Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})$
in the sum at the right-hand side of formula~(\ref{(4.2)}) 
corresponding to the original regular system equals the sum 
of all such terms $f(x_{j_1},\dots,x_{j_n})
Z_G(\Delta'_{j'_1})\cdots Z_G(\Delta'_{j'_n})$ in the sum
corresponding to the refined partition for which
$\Delta'_{j'_1}\times\cdots\times\Delta'_{j'_n}\subset
\Delta_{j_1}\times\cdots\times\Delta_{j_n}$.

By property~(vii) of the random spectral measures all products
$$
Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
$$ 
with non-zero coefficient in~(\ref{(4.2)}) are products of 
independent random variables. We had this property in mind 
when requiring the condition that the function $f$ vanishes on 
a product $\Delta_{j_1}\times\cdots\times\Delta_{j_n}$ if 
$j_l=\pm j_{l'}$ for some $l\neq l'$. This condition is 
interpreted in the literature as discarding the hyperplanes 
$x_l=x_{l'}$ and $x_l=-x_{l'}$, \ $l,l'=1,2,\dots,n$, 
$l\neq l'$, from the domain of integration. (Let us observe
that in this case, --- unlike to the definition of the original
Wiener--It\^o integrals discussed in Chapter~7, --- we omitted 
also the hyperplanes $x_l=-x_{l'}$ and not only the hyperplanes 
$x_l=x_{l'}$, $l\neq l'$, from the domain of integration.)
Property~(a) of the functions in $\bar{{\cal H}}_G^n$ 
and property~(v) of the random spectral measures imply that
$I_G(f)=\overline{I_G(f)}$, i.e. $I_G(f)$ is a real valued 
random variable for all $f\in\hat{\bar{{\cal H}}}_G^n$. 
The relation
\begin{equation}
EI_G(f)=0, \quad \textrm{for }f\in\hat{\bar{{\cal H}}}_G^n,
\quad n=1,2,\dots \label{(4.3)}
\end{equation}
also holds. Let 
$\hat{{\cal H}}_G^n={\cal H}_G^n\cap\hat{\bar{{\cal H}}}_G^n$.
If $f\in\hat{\bar{{\cal H}}}_G^n$, then  
$\textrm{Sym}\, f\in\hat{{\cal H}}_G^n$, and
\begin{equation}
I_G(f)=I_G(\textrm{Sym}\, f). \label{(4.4)}
\end{equation}
Relation~(\ref{(4.4)}) follows immediately from the observation that
$Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})=Z_G(\Delta_{\pi(j_1)})\cdots
Z_G(\Delta_{\pi(j_n)})$ for all $\pi\in\Pi_n$.
We also claim that
\begin{equation}
EI_G(f)^2\le\frac1{n!}\|f\|^2 \quad\textrm {for \ }
f\in\hat{\bar{{\cal H}}}_G^n, \label{(4.5)}
\end{equation}
and
\begin{equation}
EI_G(f)^2=\frac1{n!}\|f\|^2 \quad\textrm {for \ } f\in\hat{{\cal H}}_G^n.
\label{($4.5'$)}
\end{equation}
More generally, we claim that
\begin{eqnarray}
EI_G(f)I_G(h)=\frac1{n!}(f,g)&&=\frac1{n!}\int f(x_1,\dots,x_n)
\overline{g(x_1,\dots,x_n)}G(\,dx_1)\dots G(\,dx_n) \nonumber \\ 
&&\qquad\qquad\qquad\qquad \textrm {for \ } f,g\in\hat{{\cal H}}_G^n.
\label{($4.5''$)}
\end{eqnarray}

Because of~(\ref{(4.1)}) and~(\ref{(4.4)}) it is enough to 
check~(\ref{($4.5''$)}).

Let ${\cal D}$ be a regular system of sets in $R^\nu$,
$j_1,\dots,j_n$ and $k_1,\dots,k_n$ be indices such that
$j_l\neq\pm j_{l'}$, $k_l\neq\pm k_{l'}$ if $l\neq l'$. Then
\begin{eqnarray*}
&&EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_n})}\\
&&\qquad\qquad=\left\{
\begin{array}{l}
G(\Delta_{j_1})\cdots G(\Delta_{j_n}) \quad\textrm{ if \ }
\{j_1,\dots,j_n\}=\{k_1,\dots,k_n\}, \\
0 \quad \textrm{otherwise.}
\end{array} \right.
\end{eqnarray*}

To see the last relation one has to observe that the product on the
left-hand side can be written as a product of independent random
variables because of property~(vii) of the random spectral measures.
If $\{j_1,\dots,j_n\}\neq\{k_1,\dots,k_n\}$, then there is an index~$l$
such that either $j_l\neq\pm k_{l'}$ for all $1\le l'\le n$, or there
exists an index $l'$, $1\le l'\le n$, such that $j_l=-k_{l'}$. In the
first case $Z_G(\Delta_{j_l})$ is independent of the remaining
coordinates of the vector
$(Z_G(\Delta_{j_1}),\dots,Z_G(\Delta_{j_n}),
\overline{Z_G(\Delta_{k_1})},\dots,\overline{Z_G(\Delta_{k_n})})$,
and $EZ_G(\Delta_{j_l})=0$. Hence the expectation of the investigated
product equals zero, as we claimed. If ${j_l}=-k_{l'}$ with some index
$l'$, then a different argument is needed, since $Z_G(\Delta_{j_l})$
and $Z_G(-\Delta_{j_l})$ are not independent. In this case we can
state that since $j_p\neq\pm j_l$ if $p\neq l$, and
$k_q\neq\pm j_l$ if $q\neq l'$, the vector
$(Z_G(\Delta_{j_l}),Z_G(-\Delta_{j_l}))$ is independent of the
remaining coordinates of the above random vector. On the other hand,
the product $Z_G(\Delta_{j_l})\overline{Z_G(-\Delta_{j_l}})$
has zero expectation, since
$EZ_G(\Delta_{j_l})\overline{Z_G(-\Delta_{j_l})}
=G(\Delta_{j_l}\cap(-\Delta_{j_l}))=0$ by property~(iii) of the
random spectral measures and the relation
$\Delta_{j_l}\cap(-\Delta_{j_l})=\emptyset$. Hence the expectation
of the considered product equals zero also in this case. If
$\{j_1,\dots,j_n\}=\{k_1,\dots,k_n\}$, then
$$
EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_n})} 
=\prod\limits_{l=1}^n EZ_G(\Delta_{j_l})
\overline{Z_G(\Delta_{j_l})}=\prod\limits_{l=1}^n G(\Delta_{j_l}).
$$

Therefore for two functions $f,g\in\hat{{\cal H}}_G^n$
we may assume that they are adapted to the same regular system
${\cal D}=\{\Delta_j,\,j=\pm1,\dots,\pm N\}$, and
\begin{eqnarray*}
EI_G(f)I_G(g)&&=EI_G(f)\overline{I_G(g)}
=\left(\frac1n\right)^2\sum\sum f(x_{j_1},\dots,x_{j_n})
\overline{g(x_{k_1},\dots,x_{k_n})} \\
&&\qquad\qquad\qquad\qquad EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_n})}\\
&&=\left(\frac1{n!}\right)^2\sum f(x_{j_1},\dots,x_{j_n})
\overline{g(x_{j_1},\dots,x_{j_n})}
G(\Delta_{j_1})\cdots G(\Delta_{j_n}) \cdot n! \\
&&=\frac1{n!}\int f(x_1,\dots,x_n)
\overline{g(x_1,\dots,x_n)} G(\,dx_1)\cdots G(\,dx_n)
=\frac1{n!}(f,g).
\end{eqnarray*}

We claim that Wiener--It\^o integrals of different order are
uncorrelated. More explicitly, take two functions
$f\in\hat{\bar{{\cal H}}}^n_G$ and $f'\in\hat{\bar{{\cal H}}}^{n'}_G$
such that $n\neq n'$. Then we have
\begin{equation}
EI_G(f)I_G(f')=0 \quad \textrm{if \ }f\in \hat{\bar{{\cal H}}}^n_G, \;\;
f'\in\hat{\bar{{\cal H}}}^{n'}_G, \textrm{ and \ } n\neq n'. 
\label{(4.6)}
\end{equation}
To see this relation observe that a regular system ${\cal D}$ can be
chosen is such a way that both $f$ and $f'$ are adapted to it.
Then a similar, but simpler argument as the previous one shows that
$$
EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_{n'}})}=0
$$
for all sets of indices $\{j_1,\dots,j_n\}$ and $\{k_1,\dots,k_{n'}\}$
if $n\neq n'$, hence the sum expressing $EI_G(f)I_G(f')$ in this case
equals zero.

\medskip
We extend the definition of Wiener--It\^o integrals to a 
more general class of kernel functions with the help of 
the following Lemma~4.1. This  is a simple result, but 
unfortunately it contains several small technical 
details, and this makes its reading unpleasant.

\medskip\noindent
{\bf Lemma 4.1.} {\it The class of simple functions
$\hat{\bar{{\cal H}}}_G^n$ is dense in the (real) Hilbert space 
$\bar{{\cal H}}_G^n$, and the class of symmetric simple function 
$\hat{{\cal H}}_G^n$ is dense in the (real) Hilbert space 
${\cal H}_G^n$.} 

\medskip\noindent
{\it Proof of Lemma 4.1.}\/
It is enough to show that $\hat{\bar{{\cal H}}}_G^n$ is dense 
in the Hilbert space  $\bar{{\cal H}}_G^n$, since the second 
statement of the lemma follows from it by a standard 
symmetrization procedure.

First we reduce the result of Lemma~4.1 to a {\it Statement~A}\/ 
and then to a {\it Statement~B}. Finally we prove {\it Statement~B}. 
In {\it Statement~A}\/ we  claim that the indicator function 
$\chi_A$ of a bounded set $A\in{\cal B}^{n\nu}$ such that 
$A=-A$ can be well approximated by a function of the form 
$g=\chi_B\in\hat{\bar{{\cal H}}}_G^n$, where $\chi_B$ is the
indicator function of an appropriate set~$B$. Actually we 
formulate this statement in a more complicated form, because 
only in such a way can we reduce the statement about the 
good approximability of a general, possibly complex valued 
function~$f\in\bar{{\cal H}}_G^n$ by a function in
$g\in\hat{\bar{{\cal H}}}_G^n$ to {\it Statement~A}.

\medskip\noindent
{\it Statement A.}\/ Let $A\in{\cal B}^{n\nu}$ be a bounded, 
symmetric set, i.e. let $A=-A$. Then for any $\varepsilon>0$ 
there is a function $g\in\hat{\bar{{\cal H}}}_G^n$ such that 
$g=\chi_B$ with some set $B\in{\cal B}^{n\nu}$, i.e. $g$~is 
the indicator function of a set~$B$ such that the inequality 
$\|g-\chi_A\|<\varepsilon$ holds with the norm of the 
space $\bar{{\cal H}}_G^n$. (Here $\chi_A$ denotes the 
indicator function of the set~$A$, and we have
$\chi_A\in\bar{{\cal H}}_G^n$.) 

If $\chi_A\in\bar{{\cal H}}_G^n$, and $A_1$ is such a set for 
which the set $A$  can be written in the form $A=A_1\cup (-A_1)$, 
and the sets $A_1$ and $-A_1$ have a positive distance from 
each other, i.e. 
$\rho(A_1,-A_1)=\inf\limits_{x\in A_1,\,y\in -A_1}\rho(x,y)>\delta$,
with some $\delta>0$, where $\rho$ denotes the Euclidean 
distance in $R^{n\nu}$, then a good approximation of 
$\chi_A$ can be given with such a function 
$g=\chi_{B\cup(-B)}\in\hat{\bar{{\cal H}}}_G^n$ for which
the sets $B$ and $-B$ are separated from each other, and 
the set $B$ is close to $A_1$. More explicitly, for all 
$\varepsilon>0$ there is a set $B\in{\cal B}^{n\nu}$ such that
$B\subset A_1^{\delta/2}=\{x\colon\;\rho(x,A_1)\le\frac\delta2\}$,
$g=\chi_{B\cup(-B)}\in\hat{\bar{{\cal H}}}_G^n$, and
$G^n(A_1\,\Delta\,B)<\frac\varepsilon2$. Here $A\Delta B$ 
denotes the symmetric difference of the sets $A$ and $B$, 
and $G^n$ is the $n$-fold direct product of the spectral 
measure~$G$ on the space $R^{n\nu}$. (The above 
properties of the set~$B$ imply that the function
$g=\chi_{B\cup(-B)}\in\hat{\bar{{\cal H}}}_G^n$ satisfies 
the relation $\|g-\chi_A\|<\varepsilon$.)

\medskip
To justify the reduction of Lemma~4.1 to {\it Statement~A}\/ 
let us observe that if two functions $f_1\in\bar{{\cal H}}_G^n$ 
and $f_2\in\bar{{\cal H}}_G^n$ can be arbitrarily well 
approximated by functions from $\hat{\bar{{\cal H}}}_G^n$ 
in the norm of this space, then the same relation 
holds for any linear combination $c_1f_1+c_2f_2$ with real 
coefficients~$c_1$ and~$c_2$. (If the functions $f_i$ are
approximated by some functions $g_i\in\hat{\bar{{\cal H}}}_G^n$,
$i=1,2$, then we may assume, by applying some refinement of the
partitions if it is necessary, that the approximating functions 
$g_1$ and $g_2$ are adapted to the same regular partition.)
Hence the proof about the arbitrarily good approximability of a 
function $f\in\bar{{\cal H}}_G^n$ by functions
$g\in\hat{\bar{{\cal H}}}_G^n$ can be reduced to the proof 
about the arbitrarily good approximability of its real part
$\textrm{Re}\, f\in\bar{{\cal H}}_G^n$ and its imaginary part
$\textrm{Im}\, f\in\bar{{\cal H}}_G^n$. Moreover, since the 
real part and imaginary part of the function~$f$ can be 
arbitrarily well approximated by such real or imaginary valued 
functions from the space $\bar{{\cal H}}_G^n$ which take only 
finitely many values, the desired approximation result can be 
reduced to the case when $f$ is the indicator function of a 
set $A\in{\cal B}^{n\nu}$ such that $A=-A$ (if $f$ is real 
valued), or it takes three values, the value $i$ on a set 
$A_1\in {\cal B}^{n\nu}$, the value~$-i$ on the set $-A_1$, 
and it equals zero on $R^{n\nu}\setminus(A_1\cup(-A_1))$ 
(if $f$ is purely imaginary valued). Besides, the 
inequalities $G^n(A)<\infty$ and $G^n(A_1)<\infty$ hold. We 
may even assume that $A$ and $A_1$ are bounded sets, because 
$G^n(A)=\lim\limits_{K\to\infty}G^n(A\cap[-K,K]^{n\nu})$, and
the same argument applies for~$A_1$.

{\it Statement~A}\/ immediately implies the desired approximation 
result in the first case when $f$  is the indicator function 
of a set~$A$ such that $A=-A$. In the second case, when such a 
function~$f$ is considered that takes the values $\pm i$ and 
zero, observe that the sets $A_1=\{x\colon\; f(x)=i\}$ and
$-A_1=\{x\colon\; f(x)=-i\}$ are disjoint. Moreover, we may assume
that they have positive distance from each other, because there
are such compact sets $K_N\subset A_1$, $N=1,2,\dots$, for which
$\lim\limits_{N\to\infty} G^n(A\setminus (K_N\cup(-K_N))=0$, and 
the two disjoint compact sets $K_N$ and $-K_N$ have positive 
distance. This enables us to restrict our attention to the 
approximation of such functions $f$ for which 
$A_1=\{x\colon\; f(x)=i\}=K_N$,
$-A_1=\{x\colon\; f(x)=-i\}=-K_N$ with one of the above defined 
sets $K_N$ with a sufficiently large index $N$, and the
function $f$ disappears on the complement of the set 
$A_1\cup(-A_1)$. To get a good approximation in this case, 
take $A_1=K_N$ and apply the 
second part of {\it Statement A}\/ for the indicator 
function $\chi_A=\chi_{K_N\cup (-K_N)}$ with the choice 
$A_1=K_N$. We get that there exists a function
$g=\chi_{B\cup(-B)}\in\hat{\bar{{\cal H}}}_G^n$ such that 
$B\subset A_1^{\delta/2}$ with a number $\delta>0$ for which 
the relation $\rho(K_N,-K_N)>\delta$ holds, and 
$G^n(A_1\,\Delta\,B)<\frac\varepsilon2$. Then we define with 
the help of the above set $B$ the function 
$\bar g\in\hat{\bar{{\cal H}}}_G^n$ as 
$\bar g(x)=i$ if $x\in B$, $\bar g(x)=-i$ if $x\in-B$ and 
$\bar g(x)=0$ otherwise. The definition of the function 
$\bar g(\cdot)$ is meaningful, since $B\cap(-B)=\emptyset$,
and it yields a sufficiently good approximation of the 
function $f(\cdot)$.  

\medskip
In the next step we reduce the proof of {\it Statement~A}\/ 
to the proof of a result called {\it Statement~B}. In this 
step we show that to prove {\it Statement~A}\/ it is enough 
to prove the good approximability of some very special (and 
relatively simple) indicator functions 
$\chi_B\in\bar{{\cal H}}_G^n$ by a function 
$g\in\hat{\bar{{\cal H}}}_G^n$. 

\medskip\noindent
{\it Statement B.}\/ Let $B=D_1\times\cdots\times D_n$ be 
the direct product of bounded sets $D_j\in{{\cal B}}^\nu$ 
such that $D_j\cap(-D_j)=\emptyset$ for all $1\le j\le n$. 
Then for all $\varepsilon>0$ there is a set 
$F\subset B\cup(-B)$, $F\in{\cal B}^{n\nu}$ such that 
$\chi_F\in\hat{\bar{{\cal H}}}_G^n$, and
$\|\chi_{B\cup(-B)}-\chi_F\|\le\varepsilon$, with the norm 
of the space $\bar{{\cal H}}_G^n$.

\medskip
To deduce {\it Statement~A}\/ from {\it Statement~B}\/ let us 
first remark that we may reduce our attention to such sets~$A$ 
in {\it Statement~A}\/ for which all coordinates of the points 
in the set~$A$ are separated from the origin. More explicitly, 
we may assume the existence of a number $\eta>0$ with the 
property $A\cap K(\eta)=\emptyset$, where 
$K(\eta)=\bigcup\limits_{j=1}^n K_j(\eta)$ with 
$K_j(\eta)=\{(x_1,\dots,x_n)\colon\; x_l\in R^\nu,\;l=1,\dots,n,\;
\rho(x_j,0)\le\eta\}$. To see our right to make such a reduction
observe that the relation $G(\{0\})=0$ implies that
$\lim\limits_{\eta\to0}G^n(K(\eta))=0$, hence
$\lim\limits_{\eta\to0}G^n(A\setminus K(\eta))=G^n(A)$. At this point
we exploited a weakened form of the non-atomic property of the
spectral measure~$G$, namely the relation~$G(\{0\})=0$.

\medskip
First we formulate a result that we prove somewhat later,
and reduce the proof of {\it Statement~A}\/ to that of 
{\it Statement~B}\/ with its help. We claim that for all 
numbers~$\varepsilon>0$, $\bar\delta>0$ and bounded sets 
$A\in{\cal B}^{n\nu}$ such that $A=-A$, and
$A\cup K(\eta)=\emptyset$ there is a finite sequence of 
bounded sets $B_j\in{\cal B}^{n\nu}$, $j=\pm1,\dots,\pm N$, 
with the following properties. The sets $B_j$ are disjoint, 
$B_{-j}=-B_j$, $j=\pm1,\dots,\pm N$, each set $B_j$ can be 
written in the form $B_j=D^{(j)}_1\times\cdots\times D^{(j)}_n$ 
with $D^{(j)}_k\in{\cal B}^{\nu}$, and
$D^{(-j)}_k\cap(-D^{(j)}_k)=\emptyset$ for all 
$1\le j\le N$ and $1\le k\le n$, the diameter 
$d(B_j)=\sup\{\rho(x,y)\colon\;x,y\in B_j\}$ of the 
sets $B_j$ has the bound $d(B_j)\le\bar\delta$ for all 
$1\le j\le N$, and finally the set
$B=\bigcup\limits_{j=1}^N(B_j\cup B_{-j})$ satisfies the 
relation $G^n(A\Delta B)\le\varepsilon$. 

Indeed, since we can choose $\varepsilon>0$ arbitrarily small, 
the above result together with the application of 
{\it Statement~B}\/ for all functions $\chi_{B_j\cup(-B_j)}$, 
$1\le j\le N$, supplies an arbitrarily good approximation of 
the function $\chi_A$ by a function of the form
$\sum\limits_{j=1}^N\chi_{F_j}\in\hat{\bar{{\cal H}}}^n_G$
in the norm of the space $\bar{{\cal H}}_G^n$. Moreover, the
random variable
$\sum\limits_{j=1}^N\chi_{F_j}\in\hat{\bar{{\cal H}}}^n_G$
agrees with the indicator function of the set 
$\bigcup\limits_{j=1}^N F_j$, since the sets $B_j$, 
$j=\pm1,\dots,\pm N$, are disjoint, and 
$F_j\subset B_j\cup B_{-j}$.

If the set $A$ can be written in the form $A=A_1\cup(-A_1)$ 
such that $\rho(A_1,-A_1)>\delta$, then we can make the 
same construction with the only modification that this 
time we demand that the sets $B_j$ satisfy the relation 
$d(B_j)\le\bar\delta$ with some $\bar\delta<\frac\delta2$ 
for all $1\le j\le N$. We may assume that 
$A\cap(B_j\cup B_{-j})\neq\emptyset$ for all indices~$j$, 
since we can omit those sets $B_j\cup B_{-j}$ which do not 
have this property. Since $d(B_j)<\frac\delta2$, a set 
$B_j$ cannot intersect both $A_1$ and $-A_1$. By an 
appropriate indexation of the sets $B_j$ we have 
$B_j\subset A_1^{\delta/2}$ and $B_{-j}\subset(-A_1)^{\delta/2}$ 
for all $1\le j\le N$. Then the set 
$B=\bigcup\limits_{j=1}^N (B_j\cap F_j)$ and the function
$g=\chi_{B\cup(-B)}$ satisfy the second part of 
{\it Statement A}.

To find a sequence~$B_j$, $j=\pm1,\dots,\pm N$, for a set~$A$ 
such that $A=-A$, and $A\cup K(\eta)=\emptyset$ with the  
properties needed in the above argument observe that 
there is a sequence of finitely many bounded sets $B_j$ 
of the form $B_j=D^{(j)}_1\times\cdots\times D^{(j)}_n$,
$D^{(j)}_l\in{\cal B}^\nu$, $1\le j\le N$ with some 
$N<\infty$, whose union $B=\bigcup B_j$ satisfies the 
relation $G^n(A\,\Delta\,B)<\frac\varepsilon2$. Because of 
the symmetry property $A=-A$ of the set~$A$ we may assume 
that these sets $B_j$ have such an indexation with both
positive and negative integers for which $B_j=-B_{-j}$. 
We may also demand that $B_j\cap A\neq\emptyset$ for all 
sets~$B_j$. Besides, we may assume, by dividing the sets 
$D^{(j)}_l$ appearing in the definition of the sets~$B_j$ 
into smaller sets if this is needed that their diameter
$d(D^{(j)}_l)<\max(\frac\eta2,\frac{\bar\delta}n)$. This 
implies because of the relation $A\cap K(\eta)=\emptyset$
that $D^{(j)}_l\cap(-D^{(j)}_l)=\emptyset$ for all $j$ and 
$1\le l\le n$. The above constructed sets $B_j$ may be 
non-disjoint, but with the help of their appropriate further 
splitting and  a proper indexation of the sets obtained in 
such a way we get such a partition of the set $B$ which 
satisfies all conditions we demanded. For the sake of 
completeness we present a partition of the set $B$ with 
the properties we need.

Let us first take for all $1\le l\le n$ the following 
partition of $R^\nu$ with the help of the sets $D^{(j)}_l$,
$1\le j\le N$. For a fixed number~$l$ this partition 
consists of all sets $\bar D^{(l)}_r$ of the form 
$\bar D^{(l)}_r=\bigcap\limits_{1\le j\le N} F^{r(j)}_{l,j}$, 
where the indices $r$ are sequences $(r(1),\dots,r(N))$ 
of length~$N$ with $r(j)=1,2$ or~3, $1\le j\le N$, and
$F^{(1)}_{l,j}=D^{(j)}_l$, $F^{(2)}_{l,j}=-D^{(j)}_l$,
$F^{(3)}_{l,j}=R^\nu\setminus(D^{(j)}_l\cup(-D^{(j)}_l))$. Then 
$B$ can be represented as the union of those sets of the
form $\bar D^{(1)}_{r_1}\times\cdots\times\bar D^{(n)}_{r_n}$
which are contained in~$B$.

\medskip\noindent
{\it Proof of Statement B.} To prove {\it Statement~B}\/ 
first we show that for all $\bar\varepsilon>0$  there is a 
regular system ${\cal D}=\{\Delta_l,\,l=\pm1,\dots,\pm N\}$ 
such that all sets $D_j$ and $-D_j$, $1\le j\le n$, 
appearing in the formulation of {\it Statement~B}\/ can 
be expressed as the union of some elements $\Delta_l$ 
of ${\cal D}$, and $G(\Delta_l)\le\bar\varepsilon$ 
for all $\Delta_l\in{\cal D}$.

In a first step we prove a weakened version of this 
statement. We show that there is a regular system
$\bar{{\cal D}}=\{\Delta'_l,\, l=\pm1,\dots,\pm N'\}$ such 
that all sets $D_j$ and $-D_j$ can be expressed as the union 
of some sets $\Delta'_l$ of $\bar{{\cal D}}$. But we have no 
control on the measure~$G(\Delta'_l)$ of the elements of 
this regular system $\bar{{\cal D}}$. To get such a regular 
system we define the sets $\Delta'(\varepsilon_s,\,1\le |s|\le n)
=D^{\varepsilon_1}_1\cap(-D_1)^{\varepsilon_{-1}}
\cap\cdots\cap D^{\varepsilon_n}_n\cap(-D_n)^{\varepsilon_{-n}}$
for all vectors $(\varepsilon_s,\,1\le|s|\le n)$ such that  
$\varepsilon_s=\pm1$ for all $1\le |s|\le n$, and the vector 
$(\varepsilon_s,\,1\le |s|\le n)$ contains at least  one 
coordinate~$+1$, and $D^1=D$, $D^{-1}=R^\nu\setminus D$ for 
all sets $D\in{\cal B}^\nu$. Then taking an appropriate 
reindexation of the sets
$\Delta'(\varepsilon_s,\,1\le|s|\le n)$ we get a regular 
system $\bar{{\cal D}}$ with the desired properties. (In this
construction the sets $\Delta'(\varepsilon_s,\,1\le|s|\le n)$ 
are disjoint, and during their reindexation we drop those 
of them which equal the empty set.) To see that 
$\bar{{\cal D}}$ with a good indexation is a
regular system observe that for a set
$\Delta'_l=\Delta'(\varepsilon_s,\,1\le|s|\le n)\in\bar{{\cal D}}$ 
we have
$-\Delta'_l=\Delta'(\varepsilon_{-s},\,1\le|s|\le n)\in\bar{{\cal D}}$, 
and $\Delta'_l\cap(-\Delta'_l)\subset D_j\cap(-D_j)=\emptyset$ 
with some index $1\le j\le n$. (We had to exclude the 
possibility $\Delta_l=-\Delta_l$.)

Next we show that by appropriately refining the above regular
system $\bar{{\cal D}}$ we can get such a regular system
${\cal D}=\{\Delta_l,\,l=\pm1,\dots,\pm N\}$ which satisfies 
the additional property $G(\Delta_l)\le\bar\varepsilon$ for 
all $\Delta_l\in{\cal D}$. To show this let us observe that 
there is a finite partition
$\{E_1,\dots,E_l\}$ of $\bigcup\limits_{j=1}^n(D_j\cup(-D_j))$ 
such that $G(E_j)\le\bar\varepsilon$ for all $1\le j \le l$. 
Indeed, the closure of $D=\bigcup\limits_{j=1}^n(D_j\cup(- D_j))$ 
can be covered by open sets $H_i\subset R^\nu$ such that 
$G(H_i)\le\bar\varepsilon$ for all sets $H_i$ because of the 
non-atomic property of the measure~$G$, and by the 
Heyne--Borel theorem this covering can be chosen finite.
With the help of these sets $H_i$ we can get a partition
$\{E_1,\dots,E_l\}$ of $\bigcup\limits_{j=1}^n(D_j\cup(-D_j))$
with the desired properties.

Then we can make the following construction with the help 
of the above sets~$E_j$ and $\Delta'_l$. Take a pair of 
elements $(\Delta'_l,\Delta'_{-l})=(\Delta'_l,-\Delta'_l)$,
of $\bar{{\cal D}}$, and split up the
set $\Delta'_l$ with the help of the sets~$E_j$
to the union of finitely many disjoint sets of the form
$\Delta_{l,j}=\Delta'_l\cap E_j$. Then
$G(\Delta_{l,j})<\bar\varepsilon$ for all sets $\Delta_{l,j}$, 
and we can write the set $\Delta'_{-l}$ as the union of the
disjoint sets $-\Delta_{l,j}$. By applying this procedure for 
all pairs $(\Delta'_l,\Delta'_{-l})$ and by reindexing the sets
$\Delta_{l,j}$ obtained by this procedure in an appropriate way 
we get a regular system ${\cal D}$ with the desired properties.

To prove {\it Property~B}\/ let us write $B\cup(-B)$ as the 
union of products of sets of the form 
$\Delta_{l_1}\times\cdots\times\Delta_{l_n}$ with sets
$\Delta_{l_j}\in{\cal D}$, $1\le j\le n$. Here such a regular 
system ${\cal D}$ is considered which satisfies the properties
demanded at the start of proof of {\it Statement~B}.\/ Let us 
discard those products for which $l_j=\pm l_{j'}$ for some 
pair $(j,\,j')$, $j\neq j'$. We define the set $F$ about which 
we claim that it satisfies Property~B as the union of the 
remaining sets $\Delta_{l_1}\times\cdots\times\Delta_{l_n}$. 
Then $\chi_F\in\hat{\bar{{\cal H}}}_G^n$. Hence to prove 
that {\it Statement~B}\/ holds with this set~$F$ if 
$\bar\varepsilon>0$ is chosen sufficiently small it is enough 
to show that the sum of the terms 
$G(\Delta_{l_1})\cdots G(\Delta_{l_n})$ for which $l_j=\pm l_{j'}$
with some $j\neq j'$ is less than $n^2\bar\varepsilon M^{n-1}$, 
where $M=\max G(D_j\cup(-D_j))=2\max G(D_j)$. To see this
observe that for a fixed pair $(j,j')$, $j\neq j'$, the sum
of all products $G(\Delta_{l_1})\cdots G(\Delta_{l_n})$ such that
$l_j=l_{j'}$ can be bounded by $\bar\varepsilon M^{n-1}$, and 
the same estimate holds if summation is taken for products with 
the property $l_j=-l_{j'}$. Indeed, each term of this sum can be
bounded by $\bar\varepsilon G^{n-1}
\left(\prod\limits_{1\le p\le n,\,p\neq j}\Delta_{l_p}\right)$, 
and the events whose $G^{n-1}$ measure is considered in the
investigated sum are disjoint. Besides, their union is in the
product set $\prod\limits_{1\le p\le n,\,p\neq j}(D_p\cup D_{-p})$,
whose measure is bounded by $M^{n-1}$. Lemma 4.1 is proved.
\hfill$\qed$

\medskip
As the transformation $I_G(f)$ is a contraction from
$\hat{\bar{{\cal H}}}_G^n$ into $L_2(\Omega,{\cal A},P)$, 
it can uniquely be extended to the closure of 
$\hat{\bar{{\cal H}}}_G^n$, i.e. to $\bar{{\cal H}}_G^n$.
(Here $(\Omega,{\cal A},P)$ denotes the probability 
space where the random spectral measure $Z_G(\cdot)$ 
is defined.) At this point we exploit that if 
$f\in\hat{\bar{{\cal H}}}_G^n$, $N=1,2,\dots$, is a 
convergent sequence in the space $\bar{{\cal H}}_G^n$, 
then the sequence of random variables $I_G(f_N)$ is 
convergent in the space $L_2(\Omega,{\cal A},P)$, 
since it is a Cauchy sequence. With the help of this 
fact and Lemma~4.1 we can introduce the definition of 
Wiener--It\^o integrals in the general case when the 
integral of a function $f\in\bar{{\cal H}}_G^n$ is taken.

\medskip\noindent
{\bf Definition of Wiener--It\^o Integrals.} {\it Given 
a function $f\in\bar{{\cal H}}^n_G$ with a spectral 
measure $G$ choose a sequence of simple functions 
$f_N\in \hat{\bar{{\cal H}}}_G^n$, $N=1,2,\dots$, which 
converges to the function $f$ in the space 
$\bar{{\cal H}}^n_G$. Such a sequence exists by Lemma~4.1. 
The random variables $I_G(f_N)$ converge to a random 
variable in the $L_2$-norm of the probability space 
where these random variables are defined, and the limit 
does not depend on the choice of the sequence $f_N$ 
converging to $f$. This enables us to define the 
$n$-fold Wiener--It\^o integral with kernel function 
$f$ as
$$
\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)
=n!I_G(f)=\lim_{N\to\infty}n!I_G(f_N),
$$
where $f_N\in\hat{\bar{{\cal H}}}^n_G$, $N=1,2,\dots$, 
is a sequence of simple functions converging to the 
function $f$ in the space $\bar{{\cal H}}^n_G$.}
\index{Wiener--It\^o integral}

\medskip
The expression $I_G(f)$ is a real valued random variable for 
all $f\in\bar{{\cal H}}_G^n$, and relations~(\ref{(4.3)}), 
(\ref{(4.4)}), (\ref{(4.5)}), (\ref{($4.5'$)}),
(\ref{($4.5''$)}) and~(\ref{(4.6)}) remain valid for 
$f,\,f'\in\bar{{\cal H}}_G^n$ or $f\in{\cal H}_G^n$ instead of
$f,\,f'\in\hat{\bar{{\cal H}}}_G^n$ or $f\in\hat{{\cal H}}_G^n$.
Relations~(\ref{($4.5'$)}), and~(\ref{(4.6)}) imply that the 
transformation
$I_G\colon\; \textrm{Exp}\,{{\cal H}}_G\to L_2(\Omega,{\cal A},P)$ 
is an isometry. We shall show that also the following result holds.

\medskip\noindent
{\bf Theorem 4.2.} {\it Let a stationary Gaussian random field 
be given (discrete or generalized one), and let $Z_G$ denote 
the random spectral measure adapted to it. If we integrate 
with respect to this $Z_G$, then the transformation
$I_G\colon\; \textrm{\rm Exp}\,{{\cal H}}_G\to {\cal H}$, 
where ${\cal H}$ denotes the Hilbert space of the square 
integrable random variables measurable with respect to the 
$\sigma$-algebra generated by the random variables of the 
random spectral measure $Z_G$, is unitary. The transformation 
$(n!)^{1/2}I_G\colon\; {\cal H}_G^n\to{\cal H}_n$ is 
also unitary.}

\medskip
In the proof of Theorem~4.2 we need an identity whose proof is
postponed to the next chapter.

\medskip\noindent
{\bf Theorem 4.3. (It\^o's Formula.)} 
\index{It\^o's formula} 
{\it Let $\varphi_1,\dots,\varphi_m$, \ 
$\varphi_j\in{\cal H}_G^1$, $1\le j\le m$, be an orthonormal 
system in $L_G^2$. Let some positive integers $j_1,\dots,j_m$ 
be given, and let $j_1+\cdots+j_m=N$. Define for all 
$i=1,\dots,N$ the function $g_i$ as $g_i=\varphi_s$ for 
$j_1+\cdots+j_{s-1}<i\le j_1+\cdots+j_s$, $1\le s\le m$. (In 
particular, $g_i=\varphi_1$ for $0<i\le j_1$.) Then
\begin{eqnarray*}
&&H_{j_1}\left(\int\varphi_1(x)Z_G(\,dx)\right)\cdots
H_{j_m}\left(\int\varphi_m(x)Z_G(\,dx)\right)\\
&&\qquad=\int g_1(x_1)\cdots g_N(x_N)\,Z_G(\,dx_1)\cdots Z_G(\,dx_N)\\
&&\qquad=\int \textrm{Sym\,}[ g_1(x_1)\cdots 
g_N(x_N)]\,Z_G(\,dx_1)\cdots Z_G(\,dx_N).
\end{eqnarray*}
($H_j(x)$ denotes again the $j$-th Hermite polynomial with leading
coefficient~1.)}

\medskip\noindent
{\it Proof of Theorem 4.2.}\/ We have already seen that $I_G$ is 
an isometry. So it remains to show that it is a one to one map 
from $\textrm{\rm Exp}\,{{\cal H}}_G$ to ${\cal H}$ and from 
${\cal H}_G^n$ to ${\cal H}_n$.

The one-fold integral $I_G(f)$, $f\in{\cal H}_G^1$, agrees 
with the stochastic integral $I(f)$ defined in Chapter~3. 
Hence $I_G(e^{i(n,x)})=X(n)$ in the discrete field case, 
and $I_G(\tilde\varphi)=X(\varphi)$, $\varphi\in{\cal S}$, 
in the generalized field case. Hence 
$I_G\colon\;{\cal H}_G^1\to{\cal H}_1$ is a unitary 
transformation. Let $\varphi_1,\varphi_2,\dots$ be a
complete orthonormal basis in ${\cal H}_G^1$. Then
$\xi_j=\int\varphi_j(x)\,Z_G(\,dx)$, $j=1,2,\dots$, 
is a complete orthonormal basis in ${\cal H}_1$. 
It\^o's formula implies that for all sets of positive 
integers $(j_1,\dots,j_m)$ the random variable 
$H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)$ can be written 
as a $j_1+\cdots+j_m$-fold Wiener--It\^o integral. 
Therefore Theorem~2.1 implies that the image of 
$\textrm{Exp}\,{\cal H}_G$ is the whole space ${\cal H}$, 
and $I_G\colon\;\textrm{Exp}\,{\cal H}_G\to{\cal H}$ is
unitary.

The image of ${\cal H}_G^n$ contains ${\cal H}_n$
because of Corollary~2.4 and It\^o's formula. Since 
these images are orthogonal for different~$n$, 
formula~(\ref{(2.1)}) implies that the image of
${\cal H}_G^n$ coincides with ${\cal H}_n$. Hence
$(n!)^{1/2}I_G\colon\; {\cal H}_G^n\to{\cal H}_n$ 
is a unitary transformation. \hfill$\qed$

\medskip
The next result describes the action of shift 
transformations in ${\cal H}$. We know by Theorem~4.2 
that all $\eta\in{\cal H}$ can be written in the form
\begin{equation}
\eta=f_0+\sum_{n=1}^\infty\frac1{n!}
\int f_n(x_1,\dots,x_n)Z_G(\,dx_1)
\dots Z_G(\,dx_n) \label{(4.7)}
\end{equation}
with $f=(f_0,f_1,\dots)\in\textrm{Exp}\,{\cal H}_G$ 
in a unique way, where $Z_G$ is the random measure 
adapted to the stationary Gaussian field.

\medskip\noindent
{\bf Theorem 4.4.} {\it Let $\eta\in{\cal H}$ have the 
form~(\ref{(4.7)}). Then
$$
T_t\eta=f_0+\sum_{n=1}^\infty\frac1{n!}\int e^{i(t,x_1+\cdots+x_n)}
f_n(x_1,\dots,x_n)Z_G(\,dx_1)
\dots Z_G(\,dx_n)
$$
for all $t\in R^\nu$ in the generalized field and for all
$t\in\textrm{\BBB Z}_\nu$ in the discrete field case.}

\medskip\noindent
{\it Proof of Theorem 4.4.} Because of formulas~(\ref{(3.6)}) 
and~(\ref{($3.6'$)}) and
the definition of the shift operator~$T_t$ we have
$$
T_t\left(\int e^{i(n,x)}Z_G(\,dx)\right)=T_tX_n=X_{n+t}
=\int e^{i(t,x)}e^{i(n,x)}Z_G(\,dx), \quad t\in\textrm{\BBB Z}_\nu,
$$
and because of the identity
$\widetilde{T_t\varphi}(x)=\int e^{(i(u,x)}\varphi(u-t)\,du
=e^{i(t,x)}\tilde\varphi(x)$ for $\varphi\in{\cal S}$
\begin{eqnarray*}
T_t\left(\int\tilde\varphi(x)\,Z_G(\,dx)\right)
&=&T_tX(\varphi)=X(T_t\varphi) \\
&=&\int e^{i(t,x)}\tilde\varphi(x)Z_G(\,dx), \quad \varphi\in{\cal S},
\quad t\in R^\nu,
\end{eqnarray*}
in the discrete and generalized field cases respectively. Hence
$$
T_t\left(\int f(x) Z_G(\,dx)\right)=\int e^{i(t,x)}f(x) Z_G(\,dx) \quad
\textrm{if } f\in {\cal H}^1_G
$$
for all $t\in\textrm{\BBB Z}_\nu$ in the discrete field and for all
$t\in R^\nu$ in the generalized field case. This means that
Theorem~4.4 holds in the special case when $\eta$ is a one-fold
Wiener--It\^o integral. Let $f_1(x),\dots,f_m(x)$ be an
orthogonal system in ${\cal H}^1_G$. The set of functions
$e^{i(t,x)}f_1(x),\dots,e^{i(t,x)}f_m(x)$ is also an orthogonal
system in ${\cal H}^1_G$. ($t\in\textrm{\BBB Z}_\nu$ in the discrete and
$t\in R^\nu$ in the generalized field case.) Hence It\^o's formula
implies that Theorem~4.4 also holds for random variables of the
form
$$
\eta=H_{j_1}\left(\int f_1(x)Z_G(\,dx)\right)\cdots
H_{j_m}\left(\int f_m(x)Z_G(\,dx)\right)
$$
and for their finite linear combinations. Since these linear
combinations are dense in ${\cal H}$ Theorem~4.4 holds true.
\hfill$\qed$

\medskip
The next result is a formula for the change of variables in
Wiener--It\^o integrals.
\index{formula for the change of variables in Wiener--It\^o 
integrals}

\medskip\noindent
{\bf Theorem~4.5.} {\it Let $G$ and $G'$ be two non-atomic spectral
measures such that $G$ is absolutely continuous with respect to $G'$,
and let $g(x)$ be a complex valued function such that
\begin{eqnarray*}
g(x)&=&\overline{g(-x)}, \\
|g^2(x)|&=&\frac{dG(x)}{dG'(x)}.
\end{eqnarray*}
For every $f=(f_0,f_1,\dots)\in\textrm{\rm Exp}\,{\cal H}_G$, we define
$$
f'_n(x_1,\dots,x_n)=f_n(x_1,\dots,x_n)g(x_1)\cdots g(x_n),
\quad n=1,2,\dots,\quad f'_0=f_0.
$$
Then $f'=(f'_0,f'_1,\dots)\in \textrm{\rm Exp}\,{\cal H}_{G'}^n$, and
\begin{eqnarray*}
&&f_0+\sum_{n=1}^\infty\int\frac1{n!}f_n(x_1,\dots,x_n)
Z_G(\,dx_1)\dots Z_G(\,dx_n) \\
&&\qquad \stackrel{\Delta}{=}f'_0+\sum_{n=1}^\infty\frac1{n!}
\int f_n'(x_1,\dots,x_n)Z_{G'}(\,dx_1) \dots Z_{G'}(\,dx_n),
\end{eqnarray*}
where $Z_G$ and $Z_{G'}$ are Gaussian random spectral measures
corresponding to $G$ and~$G'$.}

\medskip\noindent
{\it Proof of Theorem 4.5.} We have $\|f'_n\|_{G'}=\|f_n\|_G$, hence
$f'\in\textrm{Exp}\,{\cal H}_{G'}$. Let $\varphi_1,\varphi_2,\dots$ be a
complete orthonormal system in ${\cal H}_G^1$. Then
$\varphi'_1,\varphi_2',\dots$, \ $\varphi'_j(x)=\varphi_j(x)g(x)$ for
all $j=1,2,\dots$ is a complete orthonormal system in ${\cal H}^1_{G'}$.
All functions $f_n\in{\cal H}_G^n$ can be written in the form
$f(x_1,\dots,x_n)=\sum c_{j_1,\dots,j_n}
\textrm{Sym}\,(\varphi_{j_1}(x_1)\cdots
\varphi_{j_n}(x_n))$. Then
$f'(x_1,\dots,x_n)=\sum c_{j_1,\dots,j_n}
\textrm{Sym(}\,\varphi'_{j_1}(x_1)\cdots
\varphi'_{j_n}(x_n))$. Rewriting all terms
$$
\int\textrm{Sym}\,(\varphi_{j_1}(x_1)\cdots\varphi_{j_n}(x_n))
Z_G(\,dx_1)\dots Z_G(,dx_n)
$$
and
$$
\int\textrm{Sym}\,(\varphi'_{j_1}(x_1)\cdots\varphi'_{j_n}(x_n))
Z_{G'}(\,dx_1)\dots Z_{G'}(,dx_n)
$$
by means of It\^o's formula we
get that $f$ and $f'$ depend on a sequence of independent standard
normal random variables in the same way. Theorem~4.5 is proved.
\hfill$\qed$

\medskip
For the sake of completeness I present in the next Lemma~4.6 
another type of change of variable result. I formulate it only in 
that simple case in which we need it in some later calculations.

\medskip\noindent
{\bf Lemma 4.6.} {\it Define for all $t>0$ the (multiplication)
transformation $T_tx=tx$ either from $R^\nu$ to $R^\nu$ or from the
torus $[-\pi,\pi)^\nu$ to the torus $[-t\pi,t\pi)^\nu$. Given a
spectral measure $G$ on $R^\nu$ or on $[-\pi,\pi)^\nu$ define the
spectral measure $G_t$ on $R^\nu$ or on $[-t\pi,t\pi)^\nu$ by the
formula $G_t(A)=G(\frac At)$ for all measurable sets~$A$, and
similarly define the function
$f_{k,t}(x_1,\dots,x_k)=f_k(tx_1,\dots,tx_k)$ for all measurable
functions~$f_k$ of $k$ variables, $k=1,2,\dots$, with
$x_j\in R^\nu$ or $x_j\in [-\pi,\pi)^\nu$ for all $1\le j\le k$,
and put $f_{0,t}=f_0$. If
$f=(f_0,f_1,\dots)\in\textrm{\rm Exp}\,{\cal H}_G$, then
$f_t=(f_{0,t},f_{1,t},\dots)\in\textrm{\rm Exp}\,{\cal H}_{G_t}$, and
\begin{eqnarray*}
&&f_0+\sum_{n=1}^\infty\int\frac1{n!}f_n(x_1,\dots,x_n)
Z_G(\,dx_1)\dots Z_G(\,dx_n) \\
&&\qquad \stackrel{\Delta}{=}f_{0,t}+\sum_{n=1}^\infty\frac1{n!}
\int f_{n,t}(x_1,\dots,x_n)Z_{G_t}(\,dx_1) \dots Z_{G_t}(\,dx_n),
\end{eqnarray*}
where $Z_G$ and $Z_{G_t}$ are Gaussian random spectral measures
corresponding to $G$ and~$G'$.}

\medskip\noindent
{\it Proof of Lemma~4.6.}\/ It is easy to see that
$f_t=(f_{0,t},f_{1,t},\dots)\in\textrm{\rm Exp}\,{\cal H}_{G_t}$. 
Moreover, we may define the random spectral measure $Z_{G_t}$ in the
identity we want to prove by the formula $Z_{G_t}(A)=Z_G(\frac At)$.
But with such a choice of $Z_{G_t}$ we can write even $=$ instead of
$\stackrel{\Delta}{=}$ in this formula. \hfill$\qed$

\medskip
The next result shows a relation between Wick polynomials and
Wiener--It\^o integrals.

\medskip\noindent
{\bf Theorem 4.7.} {\it Let a stationary Gaussian field be given, and
let $Z_G$ denote the random spectral measure adapted to it. Let
$P(x_1,\dots,x_m)=\sum c_{j_1,\dots,j_n}x_{j_1}\cdots x_{j_n}$
be a homogeneous polynomial of degree~$n$, and let
$h_1,\dots,h_m\in{\cal H}_G^1$. (Here $j_1,\dots,j_n$ are $n$ indices
such that $1\le j_l\le m$ for all $1\le l\le n$. It is possible that
$j_l=j_{l'}$ also if $l\neq l'$.) Define the random variables
$\xi_j=\int h_j(x)Z_G(\,dx)$, $j=1,2,\dots,m$, and the function
$\tilde P(u_1,\dots,u_n)=\sum c_{j_1,\dots,j_n} h_{j_1}(u_1)\cdots
h_{j_n}(u_n)$. Then
$$
\colon\!P(\xi_1,\dots,\xi_m)\!\colon=\int \tilde P(u_1,\dots,u_n)
Z_G(\,du_1)\dots Z_G(\,du_n).
$$
}

\medskip\noindent
{\it Remark.} If $P$ is a polynomial of degree $n$, then it can be
written as $P=P_1+P_2$, where $P_1$ is a homogeneous polynomial of
degree~$n$, and $P_2$ is a polynomial of degree less than~$n$.
Obviously,
$$
\colon\!P(\xi_1,\dots,\xi_m)\!\colon=\colon\!P_1(\xi_1,\dots,\xi_m)\!\colon
$$

\medskip\noindent
{\it Proof of Theorem 4.7.}\/ It is enough to show that
$$
\colon\!\xi_{j_1}\cdots\xi_{j_n}\!\colon=\int h_{j_1}(u_1)\cdots h_{j_n}(u_n)
Z_G(\,du_1)\dots Z_G(\,du_n).
$$
If $h_1,\dots,h_m\in{\cal H}_G^1$ are orthonormal, (all functions $h_l$
have norm~1, and if $l\neq l'$, then $h_l$ and $h_{l'}$ are either
orthogonal or $h_l=h_{l'}$), then this relation follows from a comparison
of Corollary~2.3 with It\^o's formula. In the general case an orthonormal
system $\bar h_1,\dots,\bar h_m$ can be found such that
$$
h_j=\sum_{k=1}^m c_{j,k}\bar h_k,\quad j=1,\dots,m
$$
with some real constants $c_{j,k}$. Set $\eta_k=\int \bar h_jZ_G(\,dx)$.
Then
\begin{eqnarray*}
\colon\!\xi_{j_1}\cdots\xi_{j_n}\!\colon
&=&\colon\! \left(\sum_{k=1}^m c_{j_1,k}\eta_k\right)
\cdots\left(\sum_{k=1}^m c_{j_n,k}\eta_k\right)\!\colon \\
&=&\sum_{k_1,\dots,k_n} c_{j_1,k_1}\cdots c_{j_n,k_n}
\colon\!\eta_{k_1}\cdots\eta_{k_n}\!\colon \\
&=&\sum_{k_1,\dots,k_n} c_{j_1,k_1}\cdots c_{j_n,k_n}
\int \bar h_{k_1}(u_1)\cdots\bar h_{k_n}(u_n)Z_G(\,du_1)\dots Z_G(du_n)\\
&=&
\int h_{j_1}(u_1)\cdots h_{j_n}(u_n)Z_G(\,du_1)\dots Z_G(du_n)
\end{eqnarray*}
as we claimed.  \hfill$\qed$

\medskip
We finish this chapter by showing how the Wiener--It\^o integral can
be defined if the spectral measure~$G$ may have atoms. We do this
although such a construction seems to have a limited importance as
in most applications the restriction that we apply the Wiener--It\^o
integral only in the case of a non-atomic spectral measure~$G$
causes no serious problem. If we try to give this definition by
modifying the original one, then we have to split up the atoms. The
simplest way we found for this splitting up, was the use
of randomization.

Let $G$ be a spectral measure on $R^\nu$, and let $Z_G$ be  a
corresponding  Gaussian spectral random measure on a probability
space $(\Omega,{\cal A},P)$. Let us define a new spectral measure
$\hat G=G\times\lambda_{[-\frac12,\frac12]}$ on $R^{\nu+1}$, where
$\lambda_{[-\frac12,\frac12]}$ denotes the uniform distribution on the
interval $[-\frac12,\frac12]$. If the probability space
$(\Omega,{\cal A},P)$ is sufficiently rich, a random spectral measure
$Z_{\hat G}$ corresponding to $\hat G$ can be defined on it in such a
way that $Z_{\hat G}(A\times [-\frac12,\frac12])=Z_G(A)$ for
all $A\in{\cal B}^\nu$. For $f\in \bar{{\cal H}}_G^n$ we define the function
$\hat f\in\bar{{\cal H}}_{\hat G}^n$ by the formula
$\hat f(y_1,\dots,y_n)=f(x_1,\dots,x_n)$ if $y_j$ is the juxtaposition
$(x_j,u_j)$, \ $x_j\in R^\nu$, $u_j\in R^1$, $j=1,2,\dots,n$. Finally
we define the Wiener--It\^o integral in the general case by the
formula
$$
\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)
=\int \hat f(y_1,\dots,y_n)Z_{\hat G}(\,dy_1)\dots Z_{\hat G}(dy_n).
$$
(What we actually have done was to introduce a virtual new
coordinate~$u$. With the help of this new coordinate we could reduce
the general case to the special case when $G$ is non-atomic.) If $G$
is a non-atomic  spectral measure, then the new definition of
Wiener--It\^o integrals coincides with the original one. It is easy to
check this fact for one-fold integrals, and then It\^o's formula
proves it for multiple integrals. It can be seen with the help of
It\^o's formula again, that all results of this chapter remain valid
for the new definition of Wiener--It\^o integrals. In particular, we
formulate the following result.

Given a stationary Gaussian field let $Z_G$ be the random spectral
measure adapted to it. All $f\in{\cal H}_G^n$ can be written in the
form
\begin{equation}
f(x_1,\dots,x_n)=\sum
c_{j_1,\dots,j_n}\varphi_{j_1}(x_1)\cdots\varphi_{j_n}(x_n) \label{(4.8)}
\end{equation}
with some functions $\varphi_j\in{\cal H}_G^1$, $j=1,2,\dots$. Define
$\xi_j=\int\varphi_j(x)Z_G(\,dx)$. If $f$ has the form~(\ref{(4.8)}), then
$$
\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)=
\sum c_{j_1,\dots,j_n}\colon\!\xi_{j_1}\cdots\xi_{j_n}\!\colon.
$$
The last identity would provide another possibility for defining
Wiener--It\^o integrals also in the case when the spectral 
measure~$G$ may have atoms.

\chapter{The Proof of It\^o's Formula: The Diagram Formula
and Some of Its Consequences}

We shall prove It\^o's formula with the help of the following

\medskip\noindent
{\bf Proposition 5.1.} {\it Let $f\in\bar{{\cal H}}_G^n$ and
$h\in\bar{{\cal H}}_G^1$. Let us define the functions
$$
f\underset{k}{\times} h(x_1,\dots,x_{k-1},x_{k+1},\dots,x_n)
=\int f(x_1,\dots,x_n)\overline{h(x_k)} G(\,dx_k), \quad k=1,\dots,n,
$$
and
$$
fh(x_1,\dots,x_{n+1})=f(x_1,\dots,x_n)h(x_{n+1}).
$$
Then $f\underset{k}{\times} h$, $k=1,\dots,n$, and $fh$ are in
$\bar{{\cal H}}_G^{n-1}$ and $\bar{{\cal H}}_G^{n+1}$
respectively, and
their norms satisfy the inequality
$\|f\underset{ k}{\times} h\|\le\|f\|\cdot\|h\|$ and
$\|fh\|\le\|f\|\cdot\|h\|$. The relation
$$
n!I_G(f)I_G(h)=(n+1)!I_G(fh)+\sum_{k=1}^n (n-1)!
I_G(f\underset{k}{\times}h)
$$
holds true.}

\medskip
We shall get Proposition 5.1 as the special case of the
diagram formula formulated in Theorem~5.3.

\medskip\noindent
{\it Remark.}\/ There is a small inaccuracy in the formulation 
of Proposition~5.1. We considered the Wiener--It\^o integral of the 
function $f\underset{k}{\times} h$ with arguments
$x_1$,\dots, $ x_{k-1}$, $x_{k+1}$,\dots, $x_n$, while we 
defined this integral for functions with arguments 
$x_1,\dots,x_{n-1}$. We can correct this inaccuracy for instance 
by reindexing the variables of $f\underset{k}{\times} h$ 
and working with the function
$$
(f\underset{k}{\times} h)'(x_1,\dots,x_{n-1})=
f\underset{k}{\times} h(x_{\alpha_k(1)},\dots,x_{\alpha_k(k-1)},
x_{\alpha_k(k+1)},\dots,x_{\alpha_k(n)})
$$ 
instead of $f\underset{k}{\times} h$, where  $\alpha_k(j)=j$ 
for $1\le j\le k-1$, and $\alpha_k(j)=j-1$ for $k+1\le j\le n$.

\medskip
We also need the following recursion formula for Hermite polynomials.

\medskip\noindent
{\bf Lemma 5.2.} {\it The identity
$$
H_n(x)=xH_{n-1}(x)-(n-1)H_{n-2}(x) \quad\textrm{for \ }n=1,2,\dots,
$$
holds with the notation $H_{-1}(x)\equiv0$.}

\medskip\noindent
{\it Proof of Lemma 5.2.}
\begin{eqnarray*}
H_n(x)=(-1)^ne^{x^2/2}\frac{d^n}{dx^n}(e^{-x^2/2})
&=&-e^{x^2/2}\frac d{dx}\left(H_{n-1}(x)e^{-x^2/2}\right)\\
&=x&H_{n-1}(x)-\frac d{dx}H_{n-1}(x).
\end{eqnarray*}
Since $\frac d{dx}H_{n-1}(x)$ is a polynomial of order $n-2$ 
with leading coefficient $n-1$ we can write
$$
\frac d{dx} H_{n-1}(x)=(n-1)H_{n-2}(x)+\sum_{j=0}^{n-3}c_jH_j(x).
$$
To complete the proof of Lemma~5.2 it remains to show that 
in the last expansion all coefficients~$c_j$ are zero. This 
follows from the orthogonality of the Hermite polynomials and 
the calculation
\begin{eqnarray*}
\int e^{-x^2/2}H_j(x)\frac d{dx}H_{n-1}(x)\,dx
&=&-\int H_{n-1}(x)\frac d{dx}(e^{-x^2/2}H_j(x))\,dx\\
&=&\int e^{-x^2/2}H_{n-1}(x)P_{j+1}(x)\,dx=0
\end{eqnarray*}
with the polynomial $P_{j+1}(x)=xH_j(x)-\frac d{dx}H_j(x)$ of
order~$j+1$ for $j\le n-3$. \hfill$\qed$

\medskip\noindent
{\it Proof of Theorem 4.3 via Proposition 5.1.}\/ We prove 
Theorem~4.3 by induction with respect to~$N$. Theorem~4.3 holds 
for $N=1$. Assume that it holds for~$N-1$. Let us define 
the functions
\begin{eqnarray*}
f(x_1,\dots,x_{N-1})&=&g_1(x_1)\cdots g_{N-1}(x_{N-1})\\
h(x)&=&g_N(x).
\end{eqnarray*}
Then
\begin{eqnarray*}
J&=&\int g_1(x_1)\cdots g_N(x_N)Z_G(\,dx_1)\dots Z_G(\,dx_N)\\
&=&N!\,I_G(fh)=(N-1)!\,I_G(f)I_G(h)-\sum_{k=1}^{N-1} (N-2)!\,
I_G(f\underset{k}{\times} h)
\end{eqnarray*}
by Proposition~5.1. We can write because of our induction 
hypothesis that
\begin{eqnarray*}
J&&=H_{j_1}\left(\int\varphi_1(x)Z_G(\,dx)\right)\cdots
H_{j_{m-1}}\left(\int\varphi_{m-1}(x)Z_G(\,dx)\right) \\
&&\qquad\qquad\qquad H_{j_m-1}\left(\int\varphi_{m}(x)Z_G(\,dx)\right)
\int\varphi_m(x)Z_G(\,dx)\\
&&\qquad-(j_m-1)H_{j_1}\left(\int\varphi_1(x)Z_G(\,dx)\right)\cdots
H_{j_{m-1}}\left(\int\varphi_{m-1}(x)Z_G(\,dx)\right)\\
&&\qquad\qquad\qquad H_{j_m-2}\left(\int\varphi_m(x)Z_G(\,dx)\right),
\end{eqnarray*}
where $H_{j_m-2}(x)=H_{-1}(x)\equiv0$ if $j_m=1$. This relation
holds, since
\begin{eqnarray*}
&&f\underset{k}{\times} h(x_1,\dots,x_{k-1},x_{k+1},\dots,x_{N-1})
=\int g_1(x_1)\cdots g_{N-1}(x_{N-1})
\overline{\varphi_m(x_k)}G(\,dx_k)\\
&&\quad \,\,=\left\{   
\begin{array}{l}
0 \quad \textrm{if \ } k\le N-j_m\\
g_1(x_1)\cdots g_{k-1}(x_{k-1})g_{k+1}(x_{k+1})\cdots 
g_{N-1}(x_{N-1}) \quad \textrm{if } N-j_m<k\le N-1.
\end{array} \right.
\end{eqnarray*}
Hence Lemma~5.2 implies that
\begin{eqnarray*}
J&&=\prod_{s=1}^{m-1}H_{j_s}
\left(\int \varphi_s(x)Z_G(\,dx)\right)
\biggl[ H_{j_m-1}\left(\int\varphi_m(x)Z_G(\,dx)\right)
\int\varphi_m(x)Z_G(\,dx) \\
&&\qquad -(j_m-1)H_{j_m-2}\left(\int\varphi_m(x)
Z_G(\,dx)\right)\biggr]
=\prod_{s=1}^m H_{j_s}\left(\int\varphi_s(x)Z_G(\,dx)\right),
\end{eqnarray*}
as claimed. \hfill$\qed$

\medskip
Let us fix some functions 
$h_1\in\bar{{\cal H}}_G^{n_1}$,\dots,
$h_m\in\bar{{\cal H}}_G^{n_m}$. In the next result, 
in the so-called diagram formula, we express the 
product $n_1!I_G(h_1)\cdots n_m!I_G(h_m)$ as the 
sum of Wiener--It\^o integrals. This result contains 
Proposition~5.1 as a special case. There is no unique 
terminology for this result in the literature.
We shall follow the notation of Dobrushin in~\cite{r7}.
We introduce a class of diagrams $\gamma$ denoted by
$\Gamma(n_1,\dots,n_k)$ and define with the help of
each diagram $\gamma$ in this class a function $h_\gamma$ 
which will be the kernel function of one of the 
Wiener--It\^o integrals taking part in the sum expressing 
the product of the Wiener--It\^o integrals we investigate.
First we define the diagrams $\gamma$ and the functions 
$h_\gamma$ corresponding to them, and then we formulate 
the diagram formula with their help. After the formulation 
of this result we present an example together with some 
figures which may help to understand better what the 
diagram formula is like.

We shall use the term diagram of order $(n_1,\dots,n_m)$ 
for an undirected graph of $n_1+\cdots+n_m$ vertices
which are indexed by the pairs of integers~$(j,l)$, 
$l=1,\dots,m$, \ $j=1,\dots,n_l$, and we shall call the 
set of vertices $(j,l)$, $1\le j\le n_l$ the $l$-th row 
of the diagram.  The diagrams of order $(n_1,\dots,n_m)$ 
are those undirected graphs with these vertices which 
have the properties that no more than one edge enters 
into  each vertex, and  edges can connect only pairs 
of vertices from different rows of a diagram, i.e. 
such vertices~$(j_1,l_1)$ and $(j_2,l_2)$ for which 
$l_1\neq l_2$. Let $\Gamma=\Gamma(n_1,\dots,n_m)$ denote 
the set of all diagrams of order $(n_1,\dots,n_m)$.
\index{diagram}

Given a diagram $\gamma\in\Gamma$ let $|\gamma|$ denote the
number of edges in~$\gamma$. Let there be given a set of 
functions $h_1\in\bar{{\cal H}}_G^{n_1}$,\dots, 
$h_m\in\bar{{\cal H}}_G^{n_m}$.  Let us denote the 
variables of the function $h_l$ by $x_{(j,l)}$ instead of 
$x_j$, i.e. let us write $h_l(x_{(1,l)},\dots,x_{(n_l,l)})$
instead of $h_l(x_1,\dots,x_{n_l})$.  Put $N=n_1+\cdots+n_m$. 
We introduce the function of $N$~variables corresponding 
to the vertices of the diagram by the formula
\begin{equation}
h(x_{(j,l)},\; l=1,\dots,m,\;j=1,\dots,n_l)=\prod_{l=1}^m
h_l(x_{(j,l)},\;j=1,\dots,n_l). \label{(5.1a)}
\end{equation}

For each diagram $\gamma\in\Gamma=\Gamma(n_1,\dots,n_m)$ 
we define the reenumeration of the indices of the function 
in~(\ref{(5.1a)}) in the following way. We enumerate the 
variables $x_{(j,l)}$ in such a way that the vertices 
into which no edges enter will have the numbers 
$1,2,\dots,N-2|\gamma|$, and the vertices connected by an
edge will have the numbers~$p$ and $p+|\gamma|$, where
$p=N-2|\gamma|+1,\dots,N-|\gamma|$. In such a way we have
defined a function $h(x_1,\dots,x_N)$ (with an enumeration 
of the indices of the variables depending on the diagram 
$\gamma$). After the definition of this function 
$h(x_1,\dots,x_N)$ we take that function of $N-|\gamma|$ 
variables which we get by replacing the arguments 
$x_{N-|\gamma|+p}$ by the arguments $-x_{N-2|\gamma|+p}$, 
$1\le p\le|\gamma|$ in the function $h(x_1,\dots,x_N)$. 
Then we define the function $h_\gamma$ appearing in the 
diagram formula by integrating this function by the 
product measure  
$\prod\limits_{p=1}^{|\gamma|}G(\,dx_{N-2|\gamma|+p})$. 

More explicitly, we write
\begin{eqnarray}
h_\gamma(x_1,\dots,x_{N-2|\gamma|})&&=\idotsint 
h(x_1,\dots,x_{N-|\gamma|},
-x_{N-2|\gamma|+1},\dots,-x_{N-|\gamma|}) \nonumber  \\
&&\qquad G(\,dx_{N-2|\gamma|+1})\dots G(\,dx_{N-|\gamma|}).
\label{(5.1)}
\end{eqnarray} 
The function $h_\gamma$ depends only on the variables
$x_1,\dots,x_{N-2|\gamma|}$, i.e. it is independent of how 
the vertices connected by edges are indexed. Indeed, it 
follows from the evenness of the spectral measure that 
by interchanging the indices $s$ and $s+\gamma$ of two 
vertices connected by an edge we do not change the value 
of the  integral $h_\gamma$. Let us now consider the
Wiener--It\^o integrals $(N-2|\gamma|)!I_G(h_\gamma)$. In the
diagram formula we shall show that the product of the
Wiener--It\^o integrals we considered  can be expressed
as the sum of these Wiener--It\^o integrals. To see that
the identity appearing in the diagram formula is 
meaningful observe that although the function $h_\gamma$ 
may depend on the numbering of those vertices 
of~$\gamma$ from which no edge starts, the function 
$\textrm{Sym}\,h_\gamma$, and therefore the Wiener--It\^o 
integral $I_G(h_\gamma)$ does not depend on it. 

Now I shall formulate the diagram formula. Then I 
make a remark about the definition of the function 
$h_\gamma$ in it and discuss an example to show 
how to calculate the terms appearing in this result.

\medskip\noindent
{\bf Theorem 5.3. (Diagram Formula.)} 
\index{diagram formula} 
{\it For all functions $h_1\in\bar{{\cal H}}_G^{n_1}$,\dots, 
$h_m\in\bar{{\cal H}}_G^{n_m}$, \ $n_1,\dots,n_m=1,2,\dots$, 
the following relations hold:

\medskip
\begin{description}
\item[(A)] $h_\gamma\in\bar{{\cal H}}_G^{N-2|\gamma|}$, and
$\|h_\gamma\|\le\prod\limits_{j=1}^m\|h_j\|$ for all 
$\gamma\in\Gamma$.

\item[(B)] $n_1!I_G(h_1)\cdots n_m! I_G(h_m)
=\sum\limits_{\gamma\in\Gamma}(N-2|\gamma|)!I_G(h_\gamma)$.
\end{description}

\medskip
Here $\Gamma=\Gamma(n_1,\dots,n_m)$, and the functions $h_\gamma$
agree with the functions $h_\gamma$ defined before the formulation of 
Theorem~5.3. In particular, $h_\gamma$ was defined in~(\ref{(5.1)}).}

\medskip\noindent
{\it Remark 1.}\/ In the special case $m=2$, $n_1=n$, $n_2=1$ 
Theorem~5.3 coincides with Proposition~5.1. To see this it 
is enough to observe that $h(-x)=\overline{h(x)}$ for all 
$h\in \bar{{\cal H}}_G^1$.

\medskip\noindent
{\it Remark 2.}\/ Observe that at the end of the definition
of the function $h_\gamma$ we replaced the variable
$x_{N-|\gamma|+p}$ by the variable $-x_{N-2|\gamma|+p}$
and not by $x_{N-2|\gamma|+p}$. This is related to the 
fact that in the Wiener--It\^o integral we integrate 
with respect a complex valued random measure $Z_G$ which 
has the property 
$EZ_G(\Delta)Z_G(-\Delta)=EZ_G(\Delta)\overline{Z_G(\Delta)}
=G(\Delta)$, while $EZ_G(\Delta)Z_G(\Delta)=0$ if 
$\Delta\cap(-\Delta)=\emptyset$. In the case of the
original Wiener--It\^o integral considered in Chapter~7
the situation is a bit different. In that case we 
integrate with respect to a real valued Gaussian 
orthonormal random measure $Z_\mu$ which has the property 
$EZ_\mu^2(\Delta)=\mu(\Delta)$. In that case a diagram 
formula also holds, but it has a slightly different form.
The main difference is that in that case we define the
function $h_\gamma$ (because of the above mentioned 
property of the random measure $Z_\mu$) by replacing
the variable $x_{N-|\gamma|+p}$ by the variable 
$x_{N-2|\gamma|+p}$.

\medskip
To make the notation in the diagram formula more 
understandable let us consider the following example. 

\medskip\noindent
{\it Example.}\/ Let us take four functions 
$h_1=h_1(x_1,x_2,x_3)\in\bar{\cal{H}}^3_G$,
$h_2=h_2(x_1,x_2)\in\bar{\cal{H}}^2_G$,
$h_3=h_3(x_1,x_2,x_3,x_4,x_5)\in\bar{\cal{H}}^5_G$ 
and $h_4=h_4(x_1,x_2,x_3,x_4)\in\bar{\cal{H}}^4_G$, 
and consider the product of Wiener--It\^o integrals 
$3!I_G(h_1)2!I_G(h_2)5!I_G(h_3)4!I_G(h_4)$. Let us
look how to calculate the kernel function $h_\gamma$ 
of a Wiener--It\^o integral 
$(14-2|\gamma|)!I_G(h_\gamma)$,
$\gamma\in\Gamma(3,2,5,4)$, appearing in the diagram
formula.

\medskip\noindent
We have to consider the class of diagrams $\Gamma(3,2,5,4)$,
i.e. the diagrams with vertices which are indexed in the first 
row as $(1,1)$, $(2,1)$, $(3,1)$, in the second row as 
$(1,2)$, $(2,2)$, in the third row in as $(1,3)$, $(2,3)$, 
$(3,3)$, $(4,3)$, $(5,3)$ and in the fourth row as $(1,4)$, 
$(2,4)$, $(3,4)$, $(4,4)$. (See Fig.~\ref{A}.)


\fig{olddiag1}{40}{The vertices of the diagrams 
$\gamma\in\Gamma(3,2,5,4)$.}{A}
 
Let us take a diagram $\gamma\in\Gamma(3,2,5,4)$, and let 
us see how we can calculate the kernel function $h_\gamma$ 
of the Wiener--It\^o integral corresponding to it. We 
also draw some pictures which may help in following this 
calculation. Let us consider for instance the diagram 
$\gamma\in\Gamma(3,2,5,4)$ with edges $((2,1),(4,3))$, 
$((3,1),(1,3))$, $((1,2),(2,4))$, $((2,2),(5,3))$, 
$((3,3),(3,4))$. Let us draw the diagram~$\gamma$ 
with its edges and with such a reenumeration of the 
vertices which helps in writing up the function 
$h(\cdot)$ (with $N=14$ variables) corresponding to 
this diagram $\gamma$ and introduced before the 
definition of the function $h_\gamma$.


The function defined in~(\ref{(5.1a)}) equals in the present 
case
\begin{eqnarray*}
&&h_1(x_{(1,1)},x_{(2,1)},x_{(3,1)}) h_2(x_{(1,2)},x_{(2,2)})
h_3(x_{(1,3)},x_{(2,3)},x_{(3,3)},x_{(4,3)},x_{(5,3)}) \\ 
&&\qquad h_4(x_{(1,4)},x_{(2,4)},x_{(3,4)},x_{(4,4)}).
\end{eqnarray*}
The variables of this function are indexed by the labels
of the vertices of~$\gamma$. We made a relabelling of
the vertices of the diagram~$\gamma$ in such a way that
by changing the indices of the above function with the
help of this relabelling we get the function $h(\cdot)$ 
corresponding to the diagram~$\gamma$. In the next step 
we shall make such a new relabelling of the vertices 
of~$\gamma$ which helps to write up the functions
$h_\gamma$ we are interested in. 
(See Fig.~\ref{B}.)
 

\fig{olddiag2}{40}{The diagram $\gamma$ we are working 
with and the reenumeration of its vertices.}{B}

The function $h(\cdot)$ (with $N=14$ variables) 
corresponding to the diagram $\gamma$ can be written 
(with the help of the labels of the vertices in the 
second diagram) as
\begin{eqnarray*}
&&h(x_1,x_2,\dots,x_{14}) \\
&&\qquad=h_1(x_1,x_5,x_6) h_2(x_7,x_8)
h_3(x_{11},x_2,x_9,x_{10},x_{13}) h_4(x_3,x_{12},x_{14},x_4).
\end{eqnarray*}

Let us change the enumeration of the vertices of the diagram
in a way which corresponds to the change of the arguments
$x_{N-|\gamma|+p}$ by the arguments $-x_{N-2|\gamma|+p}$. 
This is done in the next picture. (In this notation the 
sign $(-)$ denotes that the variable corresponding to this 
vertex is $-x_{N-2|\gamma|+p}$ and not $x_{N-2|\gamma|+p}$.
(See Fig.~\ref{C}.)

With the help of the above diagram we can write up the function 
$$
h(x_1,\dots,x_{N-|\gamma|},-x_{N-2|\gamma|+1},\dots,-x_{N-|\gamma|})
$$ 
corresponding to the diagram~$\gamma$ in a simple way. This
yields that in the present case the function $h_\gamma$ 
defined in~(\ref{(5.1)}) can be written in the form
\begin{eqnarray*}
h_\gamma(x_1,x_2,x_3,x_4)&&=\idotsint 
h_1(x_1,x_5,x_6) h_2(x_7,x_8) h_3(-x_6,x_2,x_9,-x_5,-x_8) \\ 
&&\qquad h_4(x_3,-x_7,-x_9,x_4) 
G(\,dx_5)G(\,dx_6)G(\,dx_7)G(\,dx_8) G(\,dx_9).
\end{eqnarray*}

\fig{olddiag3}{40}{The diagram applied for the calculation
of $h_\gamma$. The sign $-$ indicates that the corresponding
argument is multiplied by $-1$.}{C}

Here we integrate with respect to those variables $x_j$ whose
indices correspond to such a vertex of the last diagram from
which an edge starts.
Then the contribution of the diagram $\gamma$ to the sum at 
the right-hand side of diagram formula equals $4!I_G(h_\gamma)$ 
with this function~$h_\gamma$.

Let me remark that we had some freedom in choosing the 
enumeration of the vertices of the diagram~$\gamma$. Thus e.g. 
we could have enumerated the four vertices of the diagram from 
which no edge starts with the numbers 1, 2, 3 and 4 in an 
arbitrary order. A different indexation of these vertices would 
lead to a different function $h_\gamma$ whose Wiener--it\^o 
integral is the same. I have chosen that enumeration of the 
vertices which seemed to be the most natural for me.

\medskip
Naturally the product of two Wiener--It\^o integrals can be
similarly calculated, but the notation will be a bit simpler 
in this case. I briefly show such an example, because in 
the proof of Theorem~5.3 we shall be mainly interested 
in the product of two Wiener--It\^o integrals.

\medskip\noindent
{\it Example 2.}\/ Take two Wiener--It\^o integrals with 
kernel functions $h_1=h_1(x_1,x_2,x_3)\in\bar{\cal{H}}^3_G$ 
and $h_2=h_2(x_1,x_2,x_3,x_4,x_5)\in\bar{\cal{H}}^5_G$,
and calculate the product $3!I_G(h_1)5!I_G(h_2)$ with the
help of the diagram formula.

I shall consider only one diagram $\gamma\in\Gamma(3,5)$, and
briefly explain how to calculate the kernel function $h_\gamma$
of the Wiener--It\^o integral corresponding to it. Let us
consider for instance the diagram $\gamma\in\Gamma(3,5)$ which
contains the edges $((2,1),(3,2))$ and $((3,1),(5,2))$. Then 
the same calculation as before leads to the introduction of 
the diagram (Fig.~\ref{D}).

\fig{olddiag4}{30}{A diagram $\gamma$ with reenumerated vertices
that shows how to calculate the function $h_\gamma$.}{D}

This picture yields the following definition of the diagram 
$h_\gamma$ in the present case.
$$
h_\gamma(x_1,x_2,x_3,x_4)=\iint h_1(x_1,x_5,x_6)
h_2(x_2,x_3,-x_5,x_4,-x_6)G(\,dx_5)G(\,dx_6).
$$

\medskip\noindent
{\it Proof of Theorem~5.3.}\/ It suffices to prove 
Theorem~5.3 in the special case $m=2$. Then the case 
$m>2$ follows by induction.

We shall use the notation $n_1=n$, $n_2=m$, and we write
$x_1,\dots,x_{n+m}$ instead of
$x_{(1,1)},\dots,x_{(n,1)},x_{(1,2)}\dots,x_{(m,2)}$. It is 
clear that the function $h_\gamma$ satisfies Property~(a) 
of the classes $\bar{{\cal H}}_G^{n+m-2|\gamma|}$ defined 
in Chapter~4. We show that Part~(A) of Theorem~5.3 is a 
consequence of the Schwartz inequality. The validity of 
this inequality means in particular that the functions 
$h_\gamma$ satisfy also Property~(b) of the class of 
functions $\bar{{\cal H}}_G^{n+m-2|\gamma|}$.
 
To prove this estimate on the norm of $h_\gamma$ it is 
enough to restrict ourselves to such diagrams~$\gamma$ 
in which the vertices $(n,1)$ and $(m,2)$, \ $(n-1,1)$ and 
$(m-1,2)$,\dots, $(n-k,1)$ and $(m-k,2)$ are connected by 
edges with some $0\le k\le \min(n,m)$. In this case we can 
write
\begin{eqnarray*}
&&|h_\gamma(x_1,\dots,x_{n-k-1},x_{n+1},\dots,x_{n+m-k-1})|^2 \\
&&\qquad=\biggl|\int h_1(x_1,\dots,x_n)h_2(x_{n+1},
\dots,x_{n+m-k-1},-x_{n-k},
\dots,-x_n)\\
&&\qquad\qquad\qquad G(\,dx_{n-k})\dots G(\,dx_n)\biggr|^2 \\
&&\qquad\le \int |h_1(x_1,\dots,x_n)|^2 G(\,dx_{n-k})\dots G(\,dx_n) \\
&&\qquad\qquad\qquad
\int |h_2(x_{n+1},\dots,x_{n+m})|^2 G(\,dx_{n+m-k})\dots G(\,dx_{n+m})
\end{eqnarray*}
by the Schwartz inequality and the symmetry $G(-A)=G(A)$ of the
spectral measure~$G$. Integrating this inequality with respect to
the free variables we get Part~(A) of Theorem~5.3.

In the proof of Part~(B) first we restrict ourselves to 
the case when $h_1\in\hat{\bar{{\cal H}}}_G^n$ and 
$h_2\in\hat{\bar{{\cal H}}}_G^m$. Assume that they are 
adapted to a regular system 
${\cal D}=\{\Delta_j,\;j=\pm1,\dots,\pm N\}$ of subsets 
of $R^n$ with finite measure~$G(\Delta_j)$, 
$j=\pm1,\dots,\pm N$. We may even assume 
that all $\Delta_j\in{\cal D}$ satisfy the inequality 
$G(\Delta_j)<\varepsilon$ with some $\varepsilon>0$ to be 
chosen later, because otherwise we could split up the sets 
$\Delta_j$ into smaller ones. Let us fix a point 
$u_j\in\Delta_j$ in all sets $\Delta_j\in{\cal D}$. Put 
$K_i=\sup\limits_x|h_i(x)|$, $i=1,2$, and let $A$ be a 
cube containing all~$\Delta_j$.

We can write
\begin{eqnarray*}
I=n!I_G(h_1)m!I_G(h_2)&&={\sum}' h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})\\
&&\qquad\qquad Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_m})
\end{eqnarray*}
with the numbers $u_{j_p}\in\Delta_{j_p}$ and
$u_{k_r}\in\Delta_{k_r}$ we have fixed, where the summation in
$\sum'$ goes through all pairs $((j_1,\dots,j_n),(k_1,\dots,k_m))$,
$j_p,\,k_r\in\{\pm1,\dots,\pm N\}$, \ $p=1,\dots,n$, $r=1,\dots,m$,
such that $j_p\neq\pm j_{\bar p}$ and $k_r\neq\pm k_{\bar r}$ if
$p\neq\bar p$ or $r\neq\bar r$.

Write
\begin{eqnarray*}
I&&=\sum_{\gamma\in\Gamma}{\sum}^\gamma \,
h_1(u_{j_1},\dots,u_{j_n}) h_2(u_{k_1},\dots,u_{k_m}) \\
&&\qquad\qquad\qquad Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_m}),
\end{eqnarray*}
where $\sum^\gamma$ contains those terms of $\sum'$ for which
$j_p=k_r$ or $j_p=-k_r$ if the vertices $(1,p)$ and $(2,r)$ are
connected in $\gamma$, and $j_p\neq \pm k_r$ if $(1,p)$ and $(2,r)$
are not connected. Let us define the sets
\begin{eqnarray*}
A_1&=&A_1(\gamma)=\{p\colon\;p\in\{1,\dots,n\},\textrm{ and no
edge starts from }(p,1)\textrm{ in }\gamma\},\\
A_2&=&A_2(\gamma)=\{r\colon\;r\in\{1,\dots,m\},\textrm{ and no
edge starts from }(r,2)\textrm{ in }\gamma\}
\end{eqnarray*}
and
\begin{eqnarray*}
B=B(\gamma)&&=\{(p,r)\colon\; p\in\{1,\dots,n\},\; 
r\in\{1,\dots,m\},\\
&&\qquad\qquad (p,1) \textrm{ and } (r,2) 
\textrm{ are connected in }\gamma\}
\end{eqnarray*}
together with the map $\alpha\colon\;\{1,\dots,n\}\setminus A_1\to
\{1,\dots,m\}\setminus A_2$ defined as
\begin{equation}
\alpha(p)=r \quad \textrm{if } (p,r)\in B \quad \textrm{for all \ }
p\in\{1,\dots,n\}\setminus A_1. \label{(5.2)}
\end{equation}

Let $\Sigma^\gamma$ denote the value of the inner sum $\sum^\gamma$
for some $\gamma\in\Gamma$ in the last summation formula, and
write it in the form
$$
\Sigma^\gamma=\Sigma_1^\gamma+\Sigma_2^\gamma
$$
with
\begin{eqnarray*}
\Sigma_1^\gamma&&={\sum}^\gamma h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})
\prod_{p\in A_1}Z_G(\Delta_{j_p})\prod_{r\in A_2} Z_G(\Delta_{k_r})\\
&&\qquad\qquad\qquad\cdot
\prod_{(p,r)\in B} E\left(Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\right)
\end{eqnarray*}
and
\begin{eqnarray*}
\Sigma_2^\gamma&&={\sum}^\gamma h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})
\prod_{p\in A_1}Z_G(\Delta_{j_p})\prod_{r\in A_2} Z_G(\Delta_{k_r})\\
&&\qquad \cdot\left[\prod_{(p,r)\in B}Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})
-E\left( \prod_{(p,r)\in B} 
Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\right)\right].
\end{eqnarray*}
The random variables $\Sigma^\gamma_1$ and $\Sigma^\gamma_2$ are
real valued. To see this observe that if the sum defining these
expressions contains a term with arguments $\Delta_{j_p}$, and
$\Delta_{k_r}$, then it also contains the term with arguments
$-\Delta_{j_p}$ and $-\Delta_{k_r}$. This fact together 
with property~(v) of the random spectral measure~$Z_G$ and 
the analogous property of the functions~$h_1$  and~$h_2$ 
imply that $\Sigma^\gamma_1=\overline{\Sigma^\gamma_1}$ and
$\Sigma^\gamma_2=\overline{\Sigma^\gamma_2}$. Hence these 
random variables are real valued. As a consequence, we 
can bound $(n+m-2|\gamma|)!I_G(h_\gamma)-\Sigma_1^\gamma$ 
and $\Sigma_2^\gamma$ by means of an estimation of 
their second moment.

We are going to show that $\Sigma_1^\gamma$ is a good
approximation of $(n+m-2|\gamma|)!\,I_G(h_\gamma)$, and
$\Sigma_2^\gamma$ is negligibly small. This implies that
$(n+m-2|\gamma|)!I_G(h_\gamma)$ well 
approximates~$\Sigma^\gamma$. The proofs are based on
some simple ideas, but unfortunately their description
demands a complicated notation which makes their reading
unpleasant.

To estimate $(n+m-2|\gamma|)!I_G(h_\gamma)-\Sigma_1^\gamma$ 
we rewrite $\Sigma^\gamma_1$ as a Wiener--It\^o integral 
which can be well approximated by
$(n+m-2|\gamma|)!I_G(h_\gamma)$. To find the kernel 
function of this Wiener--It\^o integral we rewrite the 
sum defining~$\Sigma^\gamma_1$ by first fixing the 
variables $u_{j_p}$, $p\in A_1$, and $u_{k_r}$, $r\in A_2$, 
and summing up by the remaining variables, and after this 
summing by the variables fixed at the first step. We get that
\begin{eqnarray}
\Sigma_1^\gamma=\sum_{\substack{ j_p\colon\; 1\le |j_p|\le N
\textrm{ for all }p\in A_1\\
k_r\colon\; 1\le |k_r|\le N \textrm{ for all }r\in A_2}}
&&h_{\gamma,1}(j_p,\;p\in A_1,\, k_r,\, r\in A_2) \nonumber \\
&&\qquad \prod_{p\in A_1}Z_G(\Delta_{j_p})\prod_{r\in A_2} 
Z_G(\Delta_{k_r})
\label{(5.3)}
\end{eqnarray}
with a function $h_{\gamma,1}$ depending on the arguments
$j_p$, $p\in A_1$, and $k_r$, $r\in A_2$, with values
$j_p,k_r\in\{\pm1,\dots,\pm N\}$   defined with the help 
another function $h_{\gamma,2}$ described below. The 
function $h_{\gamma,2}$ also depends on the arguments 
$j_p$, $p\in A_1$, and $k_r$, $r\in A_2$, with values
$j_p,k_r\in\{\pm1,\dots,\pm N\}$. More explicitly, 
formula~(\ref{(5.3)}) holds with the function $h_{\gamma,1}$
defined as
\begin{equation}
h_{\gamma,1}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)=0  \label{(5.4a)}
\end{equation}
if the numbers in the set
$\{\pm j_p\colon\;p\in A_1\}\cup\{\pm k_r\colon\;r\in A_2\}$
are not all different, and
\begin{equation}
h_{\gamma,1}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)
=h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2) \label{(5.4b)}
\end{equation}
if all numbers $\pm j_p$, $p\in A_1$, and $\pm k_r$, 
$r\in A_2$ are different, where we define the function
$h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)$ for all
sequences $j_p$, $p\in A_1$ and $k_r$, $r\in A_2$, with
$j_p,k_r\in\{\pm1,\dots,\pm N\}$ (i.e. also in the case 
when some of the arguments~$j_p$, $p\in A_1$, or $k_r$, 
$r\in A_2$, agree) by the formula
\begin{eqnarray}
h_{\gamma,2}(j_p,\;p\in A_1,\,k_r,\,r\in A_2)
&&={\sum}^{\gamma,1} h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})
\nonumber \\
&&\qquad\cdot \prod_{(p,r)\in B}
E\left(Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\right).
\label{(5.5)}
\end{eqnarray}
The sum $\sum^{\gamma,1}$ in formula~(\ref{(5.5)}) which 
depends on the arguments $j_p$, $p\in A_1$, and $k_r$, 
$r\in A_2$, is defined in the following way. We sum up 
for such sequences $(j_1,\dots,j_n)$ and $(k_1,\dots,k_m)$ 
whose coordinates with $p\in A_1$ and $q\in A_2$ are 
fixed, and agree with the arguments $j_p$ and $k_r$ of the
function $h_{\gamma,2}$ at the left-hand side 
of~(\ref{(5.5)}) and whose coordinates with indices 
$p\in \{1,\dots,n\}\setminus A_1$ and 
$r\in \{1,\dots,m\}\setminus A_2$ satisfy the 
following conditions. 

Put 
$C=\{\pm j_p,\,p\in A_1\}\cup\{\pm k_r,\,r\in A_2\}$.
We demand that all numbers $j_p$ and $k_r$ with 
indices $p\in\{1,\dots,n\}\setminus A_1$ and 
$r\in\{1,\dots,m\}\setminus A_2$ are such that 
$j_p,k_r\in\{\pm1,\dots,\pm N\}\setminus C$. 
To formulate the remaining conditions let us write all 
numbers $r\in\{1,\dots,m\}\setminus A_2$ in the form 
$r=\alpha(p)$, $p\in\{1,\dots,n\}\setminus A_1$ with 
the map~$\alpha$ defined in~(\ref{(5.2)}). We also 
demand that only such sequences appear in the 
summation whose coordinates $k_r=k_{\alpha(p)}$ 
satisfy the condition $k_{\alpha(p)}=\pm j_p$ for all
$p\in\{1,\dots,n\}\setminus A_1$. Besides, all 
numbers $\pm j_p$, $p\in\{1,\dots,n\}\setminus A_1$, 
must be different. The summation in $\sum^{\gamma,1}$ 
is taken for all such sequences $j_p$, $p\in\{1,\dots,n\}$ 
and $k_r$, $r\in\{1,\dots,m\}$, whose coordinates with
$p\in\{1,\dots,n\}\setminus A_1$ and
$r\in\{1,\dots,m\}\setminus A_2$ satisfy the above 
conditions.

Formula~(\ref{(5.5)}) can be rewritten in a simpler form. 
To do this let us first observe that the condition 
$k_{\alpha(p)}=\pm j_p$ can be replaced by the condition 
$k_{\alpha(p)}=-j_p$ in it, and we can write 
$G(\Delta_{j_p})$ instead of the term
$EZ_G(\Delta_{j_p})Z_G(\Delta_{k_r})$ (with $(p,r)\in B$) 
in the product at the end of~(\ref{(5.5)}). This follows 
from the fact that
$EZ_G(\Delta_{j_p})Z_G(\Delta_{k_r})=EZ_G(\Delta_{j_p})^2=0$
if $k_r=j_p$ and $EZ_G(\Delta_{j_p})Z_G(\Delta_{k_r})
=EZ_G(\Delta_{j_p}Z_G(-\Delta_{j_p})=G(\Delta_{j_p})$ if 
$k_r=-j_p$. Besides, the expression in~(\ref{(5.5)}) 
does not change if we take summation for all such 
sequences for which the number $j_p$ with coordinate 
$p\in\{1,\dots,n\}\setminus A$ takes all possible
values $j_p\in\{\pm1,\dots,\pm N\}$, because in such a 
way we  only attach such terms to the sum which equal zero. 
This follows from the fact that both functions $h_1$ and 
$h_2$ are adapted to the regular system~${\cal D}$, hence
$h_1(u_{j_1},\dots,u_{j_n})h_2(u_{k_1},\dots,u_{k_m})=0$ 
if for an index 
$p\in\{1,\dots,n\}\setminus A_1$ $j_p=\pm j_{p'}$ with some
$p\neq p'$ or $j_p=-k_r$ with some $(p,r)\in B$, and the 
same relation holds if there exists some $r'\in A_2$ 
such that $j_p=\pm k_{r'}$.

The above relations enable us to rewrite~(\ref{(5.5)}) in 
the following simpler form. Let us define that map $\alpha^{-1}$ 
on the set $\{1,\dots,m\}\setminus A_2$  which is the 
inverse of the map $\alpha$ defined in~(\ref{(5.2)}), 
i.e.\ $\alpha^{-1}(r)=p$  if $(p,r)\in B$. With this 
notation we can write
\begin{eqnarray}
&&h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2) \nonumber  \\
&&\qquad= \!\!\!\!\!\!\!\!\!\!
\sum_{\substack{ j_p,\,p\in \{1,\dots,n\}\setminus A_1,\\
1\le |j_p|\le N \textrm{ for all indices } p }}
\!\! \!\!\!\!\!\!\!\!\!\!
h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_r},\,r\in A_2, -u_{j_{\alpha^{-1}(r)}},
\,r\in\{1,\dots,m\}\setminus A_2)\nonumber  \\
&&\qquad\qquad\qquad\qquad
\prod_{p\in \{1,\dots,n\}\setminus A_1} G(\Delta_{j_p}). 
\label{(5.6)}
\end{eqnarray}
Formula~(\ref{(5.6)}) can be rewritten in integral form as
\begin{eqnarray}
&&h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2) \label{(5.7)} \\
&&\qquad=\int h_1(u_{j_p},\,p\in A_1,\;
x_p,\,p\in\{1,\dots,n\}\setminus A_1) \nonumber  \\
&&\qquad\qquad h_2(u_{k_r},\,r\in A_2,\; -x_{\alpha^{-1}(r)},
\,r\in\{1,\dots,m\}\setminus A_2)
\prod_{p\in \{1,\dots,n\}\setminus A_1} G(\,dx_p). \nonumber
\end{eqnarray}

We define with the help of $h_{\gamma,1}$ and 
$h_{\gamma,2}$ two new functions on $R^{(n+m-2|\gamma|)\nu}$ 
with arguments $x_1,\dots,x_{n+m-2|\gamma}$. The first 
one will be the kernel function of the Wiener--It\^o 
integral expressing~$\Sigma^\gamma_1$, and the second 
one will be equal to the function~$h_\gamma$ defined
in~(\ref{(5.1)}). We define these functions in two 
steps. In the first step we reindex the arguments of 
the functions~$h_{1,\gamma}$ and~$h_2,\gamma$ to get 
functions depending on sequences 
$j_1,\dots,j_{n+m-2|\gamma|}$. For this goal we list 
the elements of the sets $A_1$ and $A_2$ as 
$A_1=\{p_1,\dots,p_{n-|\gamma|}\}$ with
$1\le p_1<p_2<\cdots<p_{n-|\gamma|}\le n$ and
$A_2=\{r_1,\dots,r_{m-|\gamma|}\}$ with
$1\le r_1<r_2<\cdots<r_{m-|\gamma|}\le m$, and define the  
maps $\beta_1\colon\; A_1\to\{1,\dots,n-|\gamma|\}$ and 
$\beta_2\colon\; A_2\to\{n-|\gamma|+1,\dots,n+m-2|\gamma|\}$
by the formulas $\beta_1(p_l)=l$ if $1\le l\le n-\gamma$,
$1\le l\le n-|\gamma|$, and $\beta_2(r_l)=l+n-|\gamma|$,
$1\le l\le m-|\gamma|$, if 
$n-|\gamma|+1\le l\le n+m-2|\gamma|$. We define with the 
help of the maps~$\beta_1$  and~$\beta_2$ the functions
$$
h_{\gamma,3}(j_1,\dots,j_{n+m-2|\gamma|})=
h_{\gamma,1}(j_{\beta_1(r_1)},\dots,j_{\beta_1(n-|\gamma|))},
k_{\beta_2(1)},\dots,k_{\beta_2(m-|\gamma|)})
$$
and
$$
h_{\gamma,4}(j_1,\dots,j_{n+m-2|\gamma|})=
h_{\gamma,2}(j_{\beta_1(r_1)},\dots,j_{\beta_1(n-|\gamma|))},
k_{\beta_2(1)},\dots,k_{\beta_2(m-|\gamma|)}),
$$
where the arguments of the functions $h_{\gamma,3}$ and
$h_{\gamma,4}$ are sequences $j_1,\dots,j_{n+m-2|\gamma|}$ 
with $j_s\in\{\pm1,\dots,\pm N\}$ for all 
$1\le s\le n+m-2|\gamma|$.

With the help of the above functions we define the 
following functions $h_{\gamma,5}$ and $h_{\gamma,6}$ 
on $R^{(n+m-2|\gamma|)\nu}$.
$$
h_{\gamma,5}(x_1,\dots,x_{n+m-2|\gamma|})=\left\{
\begin{array}{l}
h_{\gamma,3}(j_1,\dots,j_{n+m-2|\gamma|})
\quad\textrm{ if } x_l\in \Delta_{j_l}, \\
\qquad \textrm{ for all } 1\le l\le n+m-2|\gamma|\\
0 \quad\textrm{otherwise,}
\end{array} \right.
$$
and
$$
h_{\gamma,6}(x_1,\dots,x_{n+m-2|\gamma|})=\left\{
\begin{array}{l}
h_{\gamma,4}(j_1,\dots,j_{n+m-2|\gamma|})
\quad\textrm{ if } x_l\in \Delta_{j_l}, \\
\qquad \textrm{ for all } 1\le l\le n+m-2|\gamma|\\
0 \quad\textrm{otherwise.}
\end{array} \right.
$$

It follows from relation~(\ref{(5.4a)}) and the 
definition of the function $h_{\gamma,5}$ (with 
the help of the definition of  the functions 
$h_{\gamma,1}$ and $h_{\gamma,3}$) that 
$h_{\gamma,5}\in\hat{\bar{{\cal H}}}_G^n$, 
and it is adapted to the regular system~${\cal D}$. 
Then relations~(\ref{(5.3)}) and the  definition of 
$h_{\gamma,5}$ also imply that
$\Sigma^\gamma_1=(n+m-2|\gamma|)!I_G(h_{\gamma,5})$.

On the other hand, I claim that the function $h_\gamma$ 
defined in~(\ref{(5.1)}) satisfies the identity 
$h_\gamma=h_{\gamma,6}$. At this point we must be a
bit more careful, because we did not define the 
function $h_\gamma$ in a unique way. The function
we get by permuting the indices of its variables 
could be also considered as the function $h_\gamma$.
This causes no problem, because we are interested
not in the function $h_\gamma$ itself but in the
expression $I_G(h_\gamma)$ which does not change
if we reindex the variables of the function $h_\gamma$.
We shall define the function $h_\gamma$ with a
special (natural) indexation of its variables, and
we prove the identity $h_\gamma=h_{\gamma,6}$ for
this function.
 
To prove the desired identity first we recall the 
definition of that function $h(x_1,\dots,x_{n+m})$ 
(depending on the diagram $\gamma$) which was 
applied in the definition of $h_\gamma$. Here we 
shall use a notation similar to that applied in 
the definition of the function~$h_{\gamma,6}$.

Put $A_1=\{p_1,\dots,p_{n-|\gamma|}\}$, 
$1\le p_1<p_2<\cdots<p_{n-|\gamma|}$,
$$
\{1,\dots,n\}\setminus A_1=\{q_1,\dots,q_{|\gamma|}\},
\quad 1\le q_1<q_2<\dots<q_{|\gamma|},
$$
and $A_2=\{r_1,\dots,r_{m-|\gamma|}\}$, 
$1\le r_1<r_2<\cdots<r_{m-|\gamma|}$,
$$
\{1,\dots,m\}\setminus A_2=\{q'_1,\dots,q'_{|\gamma|}\},
\quad 1\le q'_1<q'_2<\dots<q'_{|\gamma|},
$$ 
and define with their help the following functions. Define
the function $\beta(\cdot)$ on the set $\{1,\dots,n\}$
as $\beta(k)=s$ if $k=p_s\in A_1$, and 
$\beta(k)=n+m-2|\gamma|+t$ if 
$k=l_t\in\{1,\dots,n\}\setminus A_1$.
Define similarly the function $\delta(\cdot)$
on the set $\{n+1,\dots,n+m\}$ as
$\delta(k)=s+|\gamma|$ if $k-n=q_s\in A_2$. If 
$k-n=l'_t\in\{1,\dots,m\}\setminus A_2$, then there 
is an edge $(l_t,l'_t)\in B$ of the diagram $\gamma$,
and we define
$\delta(k)=n-|\gamma|+t$ with the index $t$ of
the number $l_t$ in this case. 

With the help of the above notations we can define the 
function $h(x_1,\dots,x_{n+m})$ which takes part in the 
definition of $h_\gamma$ in formula~(\ref{(5.1)}) as
$$
h(x_1,\dots,x_{n+m})=h_1(x_{\beta(1)},\dots,x_{\beta(n)})
h_2(x_{\delta(n+1)},\dots,x_{\delta(n+m)}).
$$

To define the kernel function of the integral in~(\ref{(5.1)})
observe that the set $\{\delta(n+1),\dots,\delta(n+m)\}$
agrees with the set $\{n-|\gamma|+1,\dots, n+m-2|\gamma|\}\cup
\{n+m-|\gamma|+1,\dots,n+m\}=C_1\cup C_2$. Put 
$\bar\delta(k)=\delta(k)$ if $\delta(k)\in C_1$ and
$\bar\delta(k)=\delta(k)-|\gamma|$ if $\delta(k)\in C_2$.
Let us also introduce $\varepsilon(j)=1$ if 
$n-|\gamma+1\le j\le n+m-2|\gamma|$, and $\varepsilon(j)=-1$ 
if $n+m-2|\gamma|+1\le j\le n+m-|\gamma|$. With such a 
notation we can write
\begin{eqnarray*}
&&h(x_1,\dots,x_{n+m-|\gamma|},-x_{n+m-2|\gamma|+1},
\dots,-x_{n+m-|\gamma|}) \\
&&\qquad=h_1(x_{\beta(1)},\dots,x_{\beta(n)})
h_2(\varepsilon(\bar \delta(n+1))x_{\bar\delta(n+1)},\dots,
\varepsilon(\bar\delta(n+m))x_{\bar\delta(n+m)})
\end{eqnarray*}
as the kernel function in the integral~(\ref{(5.1)})
defining the function $h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})$
in the present case.

By formula~(\ref{(5.1)}) we can calculate the function
$h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})$ by integrating
the above defined function
$h(x_1,\dots,x_{n+m-|\gamma|},-x_{n+m-2|\gamma|+1},
\dots,-x_{n+m-|\gamma|})$ with respect to the measure
$G(\,dx_{N-2|\gamma|+1})\dots G(\,dx_{N-|\gamma|})$.
By comparing this formula with the definition of the 
function $h_{\gamma,2}$ defined in~(\ref{(5.7)}) together
with the definition of the functions $h_{\gamma,4}$ and
$h_{\gamma,6}$ with its help one can see that
the identity $h_{\gamma,6}=h_\gamma$ holds.

We want to compare $I_G(h_{\gamma,5})$ with 
$I_G(h_{\gamma,6})$. For this goal we have to
understand where the functions $h_{\gamma,5}$
and $h_{\gamma,6}$ differ. These are those points
$(x_1,\dots,x_{n+m-2|\gamma|})$ where the function 
$h_{\gamma,5}$ disappears while the function 
$h_{\gamma,6}$ does not disappear. Observe that 
in such points $(x_1,\dots,x_{n+m-2|\gamma|})$ where 
$x_l\in\Delta_{j_l}$, $1\le l\le n+m-2|\gamma|$, with 
such indices $j_l$ for which the numbers~$\pm j_l$, 
$1\le l\le n+m-2|\gamma|$, are not all different 
the function $h_{\gamma,5}$ disappears, while the 
function $h_{\gamma,6}$ may not disappear.
But even the function $h_{\gamma,6}$ disappears if 
some of the numbers in the set 
$\{\pm j_1,\dots,\pm j_{n-|\gamma|}\}$ or in the 
set $\{\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}\}$
agree. This fact together with the identity 
$h_\gamma=h_{\gamma,6}$ and the relation between 
the functions~$h_{\gamma,5}$ and~$h_{\gamma,6}$ (implied
by the definition of the function $h_{\gamma,1}$ 
in formulas~(\ref{(5.4a)}) and~(\ref{(5.4b)})) yield 
the identity
$$
h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})=
h_{\gamma,5}(x_1,\dots,x_{n+m-2|\gamma|})
+h_{\gamma,7}(x_1,\dots,x_{n+m-2|\gamma|})
$$
with
\begin{eqnarray*}
&&h_{\gamma,7}(x_1,\dots,x_{n+m-2|\gamma|}) \\
&&\qquad =\left\{
\begin{array}{l}
h_\gamma(x_1,\dots,x_{n+m-2|\gamma|}) \quad 
\textrm{ if there exist indices }j_l, \;1\le |j_l|\le N,\\
\qquad 1\le l\le n+m-2|\gamma| \textrm{ such that } 
x_l\in \Delta_{j_l}, \; 1\le l\le n+m-2|\gamma|,\\
\qquad\textrm{ all numbers } \pm j_1,\dots,\pm j_{n-2|\gamma|}
\textrm{ are different,}\\
\qquad \textrm{ all numbers }\pm j_{n-|\gamma|+1},
\dots,\pm j_{n+m-2|\gamma|} \textrm{ are different,} \\
\qquad \textrm{ and }
\{\pm j_1,\dots,\pm j_{n-|\gamma|}\}\cap\{\pm j_{n-|\gamma|+1},
\dots,\pm j_{n+m-2|\gamma|}\}\neq\emptyset \\
0 \quad \textrm{otherwise.}
\end{array} \right.
\end{eqnarray*}

Since $\Sigma^\gamma_1=(n+m-2|\gamma|)!I_G(h_{\gamma,5})$, we have
$$
(n+m-2|\gamma|)!I_G(h_{\gamma})-\Sigma^\gamma_1
=(n+m-2|\gamma|)!I_G(h_{\gamma,7}),
$$
and
$$
E(\Sigma^\gamma_1-(n+m-2|\gamma|)!I_G(h_\gamma))^2
\le(n+m-2|\gamma|)!\|h_{\gamma,7}(\cdot)\|^2
$$
with the norm $\|\cdot\|$ in 
$\bar{{\cal H}}_G^{n+m-2|\gamma|}$.

On the other hand,
$$
\sup |h_{\gamma,7}(x_1,\dots,x_{n+m-2|\gamma|})|
\le\sup |h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})|
\le K_1 K_2 L^{|\gamma|},
$$
with $K_1=\sup|h_1|$, $K_2=\sup |h_2|$, and
$L=G(A)$, where $A$ is a fixed cube containing all $\Delta_j$. Hence
\begin{eqnarray}
E(\Sigma^\gamma_1-(n+m-2|\gamma|)!I_G(h_\gamma))^2
&\le& C_1\|h_{\gamma,7}\|^2  \nonumber \\
&\le& C_2{\sum}''G(\Delta_{j_1})\cdots 
G(\Delta_{j_{n+m-2|\gamma|}})
\nonumber \\
&\le& C\sup_j G(\Delta_j)\le C\varepsilon, \label{(5.8)}
\end{eqnarray}
where the summation $\sum''$ goes for such sequences
$j_1,\dots,j_{n+m-2|\gamma|}$, $1\le |j_l|\le N$ for all
$1\le l\le n+m-2|\gamma|$,  for which all numbers
$\pm j_1,\dots,\pm j_{n-|\gamma|}$ are different, the same relation
holds for the elements of the sequence
$\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}$, and
$$
\{\pm j_1,\dots,\pm j_{n-|\gamma|}\}\cap
\{\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}\}\neq\emptyset.
$$
The constants $C_1$, $C_2$ and $C$ may depend on the 
functions~$h_1$, $h_2$ and spectral measure~$G$, but they do not 
depend on the regular system~${\cal D}$, hence in particular on 
the parameter~$\varepsilon$. In the verification of the last
inequality in~(\ref{(5.8)}) we can exploit that each term in 
the sum $\sum''$ is a product which contains a 
factor~$G(\Delta_j)^2\le\varepsilon G(\Delta_j)$. Here an 
argument can be applied which is similar to the closing 
step in the proof of Lemma~4.1, to the final argument in the
proof of {\it Statement~B}.

\medskip
Now we turn to the estimation of $E(\Sigma^\gamma_2)^2$. 
It can be expressed as a linear combination of terms of 
the form
\begin{eqnarray}
&&\Sigma^\gamma_3(j_p,k_r,j_{\bar p},k_{\bar r},
\;p,\bar p\in\{1,\dots,n\},
\; r,\bar r\in\{1,\dots,m\}) \label{(5.9)} \\
&&\qquad =E\Biggl(\left(\prod_{p\in A_1}Z_G(\Delta_{j_p})
\prod_{r\in A_2}Z_G(\Delta_{k_r})
\prod_{\bar p\in A_1}Z_G(\Delta_{j_{\bar p}})
\prod_{\bar r\in A_2}Z_G(\Delta_{k_{\bar r}})\right)\nonumber  \\
&&\qquad\qquad\qquad\left[\prod_{(p,r)\in B} 
Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})
-E\prod_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\right]
\nonumber \\
&&\qquad\qquad\qquad\left[\prod_{(\bar p,\bar r)\in B} 
Z_G(\Delta_{j_{\bar p}}) Z_G(\Delta_{k_{\bar r}})
-E\prod_{(\bar p,\bar r)\in B} Z_G(\Delta_{j_{\bar p}})
Z_G(\Delta_{k_{\bar r}})\right]\Biggr), \nonumber
\end{eqnarray}
where $\Sigma^\gamma_3$ depends on such sequences of 
numbers $j_p,\,k_r,\,j_{\bar p},\,k_{\bar r}$ with indices 
$1\le p,\bar p\le n$ and $1\le r,\bar r\le m$  for which
$j_p,k_r,j_{\bar p},k_{\bar r}\in\{\pm1,\dots,\pm N\}$ 
for all indices $p,r,\bar p$ and $\bar r$,  $j_p=k_r$ or 
$j_p=-k_r$ if $(p,r)\in B$, otherwise all numbers 
$\pm j_p$, $\pm k_r$ are different, and the same relations 
hold for the indices~$j_{\bar p}$ and~$k_{\bar r}$ if 
the indices $p$ and $r$ are replaced by $\bar p$ 
and~$\bar r$. Moreover, the absolute value of all 
coefficients in this linear combination which depend on 
the functions $h_1$ and $h_2$ is bounded by 
$\sup|h_1(x)|^2 \sup |h_2(x)|^2$.

We want to show that for most sets of arguments
$(j_p,\,k_r,\,j_{\bar p},\,k_{\bar r})$ the expression 
$\Sigma^\gamma_3$ equals zero, and it is also small in 
the remaining cases.

Let us fix a sequence of arguments 
$j_p,\,k_r,\,j_{\bar p},\,k_{\bar r}$
of $\Sigma^\gamma_3$, and let us estimate its value with 
these arguments. Define the sets
$$
{\cal A}=\{j_p\colon\; p\in A_1\}\cup \{k_r\colon\; r\in A_2\}
\textrm{ \ \ and \ \ }
\bar{{\cal A}}=\{j_{\bar p}\colon\; \bar p\in A_1\}\cup
\{k_{\bar r}\colon\; \bar r\in A_2\}.
$$
We claim that $\Sigma^\gamma_3$ equals zero if
$\bar{{\cal A}}\neq-{\cal A}$. 

In this case there exists an index $l\in{\cal A}$ such that 
$-l\notin\bar{{\cal A}}$. Let us carry out the multiplication 
in~(\ref{(5.9)}). Because of the independence properties of 
random spectral measures each product in this expression can 
be written as the product of independent factors, and the 
independent factor containing the term $Z_G(\Delta_l)$ has
zero expectation. To see this observe that the set $\Delta_l$
appears exactly once among the arguments of the terms
$Z_G(\Delta_{j_p})$ and $Z_G(\Delta_{k_r})$, and none of these
terms contains the argument $-\Delta_l=\Delta_{-l}$. Although
$-l\notin\bar{{\cal A}}$, it may happen that $l\in\bar{{\cal A}}$. 
In this case the product under investigation contains the 
independent factor $Z_G(\Delta_l)^2$ with $EZ_G(\Delta_l)^2=0$. 
If $l\notin\bar{{\cal A}}$, then there are two possibilities. 
Either this product contains an independent factor of the form 
$Z_G(\Delta_l)$ with $EZ_G(\Delta_l)=0$, or there is a pair 
$(\bar p,\bar r)\in B$ such that 
$(j_{\bar p},k_{\bar r})=(\pm l,\pm l)$, and an
independent factor of the form
$Z_G(\Delta_l)Z_G(\pm\Delta_{-l})Z_G(\pm\Delta_l)$ with the 
property $EZ_G(\Delta_l)Z_G(\pm\Delta_{-l})Z_G(\pm\Delta_l)=0$ 
appears. Hence $\Sigma_3^\gamma=0$ also in this case.

Let
$$
{\cal F}=\bigcup_{(p,r)\in B}\{j_p,k_r\}\textrm{ \ \ and \ \ }
\bar{{\cal F}}=\bigcup_{(\bar p,\bar r)\in B}
\{(j_{\bar p},k_{\bar r}\}.
$$
A factorization argument shows again that the expression 
in~(\ref{(5.9)}) equals zero if the sets ${\cal F}\cup(-{\cal F)}$ 
and $\bar{{\cal F}}\cup(-\bar{{\cal F}})$ are  disjoint. 

In the proof of this statement we can restrict ourselves to 
the case when ${\cal A}=-\bar{{\cal A}}$. In this case 
$\pm{\cal A}$ is disjoint both of ${\cal F}\cup(-{\cal F})$ 
and $\bar{{\cal F}}\cup(-\bar{{\cal F}})$, and the product 
under investigation contains the independent factor
$\prod\limits_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})
-E\prod\limits_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})$ 
with expectation zero.

Finally in the remaining cases when ${\cal F}\cup(-{\cal F})$ 
and $\bar{{\cal F}}\cup(-\bar{{\cal F}})$ are not  disjoint, 
and ${\cal A}=-\bar{{\cal A}}$ the absolute value of the 
expression in~(\ref{(5.9)}) can be estimated from above by
\begin{equation}
C\,\varepsilon \prod G(\Delta_{j_p}) G(\Delta_{k_r}) 
G(\Delta_{j_{\bar p}})G(\Delta_{k_{\bar r}}) \label{(5.10)}
\end{equation}
with a universal constant $C<\infty$ depending only on the 
parameters~$n$ and~$m$, where the indices 
$j_p,\,k_r,\,j_{\bar p},\,k_{\bar r}$ are the same as 
in~(\ref{(5.9)}) with the following difference: All indices 
appear in~(\ref{(5.10)}) with multiplicity~1, and if both 
indices $l$ and $-l$ are present in~(\ref{(5.9)}), then one 
of them is omitted form~(\ref{(5.10)}). Moreover, for all
$j_p$ one of terms $G(\Delta_{\pm j_p})$ really appears in 
this product, and the analogous statement also holds for all
indices $k_r$, $j_{\bar p}$ and $k_{\bar r}$. The multiplying
term~$\varepsilon$ appears in~(\ref{(5.10)}), since by 
carrying out the multiplications in~(\ref{(5.9)}) and 
factorizing each term, we get that all non-zero terms have 
a factor either of the form
\begin{eqnarray*}
&&EZ_G(\Delta)^2Z_G(-\Delta)^2
=E(\,\textrm{Re}\, 
Z_G(\Delta)^2+\textrm{Im}\, Z_G(\Delta)^2)^2 \\
&&\qquad =E\,\textrm{Re}\, Z_G(\Delta)^4+E\,
\textrm{Im}\, Z_G(\Delta)^4
+2E\,\textrm{Re}\, Z_G(\Delta)^2E\,\textrm{Im}\, 
Z_G(\Delta)^2=8G(\Delta)^2
\end{eqnarray*}
or of the form $\left(E|Z_G(\Delta)|^2\right)^2=G(\Delta)^2$, 
and  $G(\Delta)<\varepsilon$ for all $\Delta\in{\cal D}$. 
(We did not mention the possibility of an independent factor 
of the form $EZ_G(\Delta)^4$ or $EZ_G(\Delta)^3Z_G(-\Delta)$ 
with $\Delta\in{\cal D}$, because as some calculation shows, 
$EZ_G(\Delta)^4=0$ and $EZ_G(\Delta)^3Z_G(-\Delta)=0$.)

Let us express $E(\Sigma_2^\gamma)^2$ as the linear 
combination of the quantities~$\Sigma^\gamma_3$, and let 
us bound each term $\Sigma^\gamma_3$ in the above way. 
This supplies an upper bound for $E(\Sigma_2^\gamma)^2$ 
by means of a sum of terms of the form~(\ref{(5.10)}). 
Moreover, some consideration shows that each of these 
terms appears only with a multiplicity less than~$C(n,m)$ 
with an appropriate constant~$C(n,m)$. Hence we can write
$$
E(\Sigma_2^\gamma)^2\le K_1^2K_2^2 C(n,m)
C\varepsilon\sum_{r=1}^{n+m}{\sum_{j_1,\dots,j_r}\!\!}''' 
G(\Delta_{j_1})\cdots G(\Delta_{j_r}),
$$
where the indices $j_1,\dots,j_r\in\{\pm1,\dots,\pm N\}$ 
in the sum $\sum'''$ are all different, and 
$K_j=\sup |h_j(x)|$, $j=1,2$. Hence
$$
E(\Sigma_2^\gamma)^2\le C_1\varepsilon 
\sum_{r=1}^{n+m} G(A)^r\le C_2\varepsilon
$$
with some appropriate constants $C_1$ and $C_2$. 
Because of the inequality~(\ref{(5.8)}), the identity
$n!I_G(h_1)m!I_G(h_2)=\sum\limits_{\gamma\in\Gamma}
(\Sigma^\gamma_1+\Sigma_\gamma^2)$ and the last 
relation the inequality
\begin{eqnarray*}
&&E\left(n!I_G(h_1)m!I_G(h_2)
-\sum_{\gamma\in\Gamma}(n+m-2|\gamma|)!
I_G(h_\gamma)\right)^2\\
&&\qquad= E\left(\sum_{\gamma\in\Gamma}
\left(\Sigma^\gamma_1+\Sigma^\gamma_2-(n+m-2|\gamma|)!
\,I_G(h_\gamma)\right)\right)^2\\
&&\qquad \le C_3\left(\sum_{\gamma\in\Gamma}
E((m+n-2|\gamma|)!\,I_G(h_\gamma)-\Sigma_1^\gamma)^2
+E(\Sigma_2^\gamma)^2\right)\le C_4\varepsilon
\end{eqnarray*}
holds. Since $\varepsilon>0$ can be chosen arbitrarily 
small, Part~B is proved in the special case 
$h_1\in\hat{\bar{{\cal H}}}_G^n$,
$h_2\in\hat{\bar{{\cal H}}}_G^m$.

If $h_1\in\bar{{\cal H}}_G^n$ and $h_2\in\bar{{\cal H}}_G^m$, 
then let us choose a sequence of functions 
$h_{1,r}\in\hat{\bar{{\cal H}}}_G^n$ and 
$h_{2,r}\in\hat{\bar{{\cal H}}}_G^m$ such that 
$h_{1,r}\to h_1$ and $h_{2,r}\to h_2$ in the norm of the 
spaces $\bar{{\cal H}}_G^n$ and $\bar{{\cal H}}_G^m$ 
respectively. Define the functions $\hat h_\gamma(r)$ and 
$h_\gamma(r)$ in the same way as $h_\gamma$, but substitute 
the pair of functions $(h_1,h_2)$ by $(h_{1,r},h_2)$
and $(h_{1,r},h_{2,r})$ in their definition. We shall 
show with the help of Part~(A) that
$$
E|I_G(h_1)I_G(h_2)-I_G(h_{1,r})I_G(h_{2,r})|\to0,
$$
and
$$
E|I_G(h_\gamma)-I_G(h_\gamma(r))|\to0
\quad\textrm{for all }\gamma\in\Gamma
$$
as $r\to\infty$. Then a simple limiting procedure shows that
Theorem~5.3 holds for all $h_1\in\bar{{\cal H}}_G^n$ and
$h_2\in\bar{{\cal H}}_G^m$.

We have
\begin{eqnarray*}
&&E|I_G(h_1)I_G(h_2)-I_G(h_{1,r})I_G(h_{2,r})|\\
&&\qquad \le E|(I_G(h_1-h_{1,r}))I_G(h_2)|
+E|I_G(h_{1,r})I_G(h_2-h_{2,r})|\\
&&\qquad \le\frac1{n!\,m!}
\left(\|h_1-h_{1,r}\|^{1/2}\|h_2\|^{1/2}
+\|h_2-h_{2,r}\|^{1/2}\|h_{1,r}\|\right)\to0,
\end{eqnarray*}
and by Part~(A) of Theorem~5.3
\begin{eqnarray*}
&&E|I_G(h_\gamma)-I_G(h_\gamma(r))|\le
E|I_G(h_\gamma)-I_G(\hat h_\gamma(r))|+
E|I_G(h_\gamma(r))-I_G(\hat h_\gamma(r))|\\
&&\qquad \le\|h_\gamma-\hat h_\gamma(r)\|^{1/2}
+\|h_\gamma(r)-\hat h_\gamma(r)\|^{1/2} \\
&&\qquad \le\|h_1-\hat h_{1,r}\|^{1/2}\|h_2\|^{1/2}
+\|h_2-\hat h_{2,r}\|^{1/2}\|h_{1,r}\|^{1/2}\to0.
\end{eqnarray*}
Theorem~5.3 is proved. \hfill$\qed$

\medskip
We formulate some consequences of Theorem~5.3. Let
$\bar\Gamma\subset\Gamma$ denote the set of complete 
diagrams, i.e.\ let a diagram $\gamma\in\bar\Gamma$  
if an edge enters in each vertex of~$\gamma$. 
\index{complete diagram}
We have $EI(h_\gamma)=0$ for all
$\gamma\in\Gamma\setminus\bar\Gamma$, since~(\ref{(4.3)}) 
holds for all $f\in\bar{{\cal H}}_G^n$, $n\ge1$. If 
$\gamma\in\bar\Gamma$, then $I(h_\gamma)\in\bar{{\cal H}}_G^0$. 
Let $h_\gamma$ denote the value of $I(h_\gamma)$ in this case. 
Now we have the following

\medskip\noindent
{\bf Corollary 5.4.} {\it For all 
$h_1\in\bar{{\cal H}}_G^{n_1}$,\dots, 
$h_n\in\bar{{\cal H}}_G^{n_m}$
$$
En_1!I_G(h_1)\cdots n_m!I_G(h_m)=
\sum_{\gamma\in\bar\Gamma}h_\gamma.
$$
(The sum on the right-hand side equals zero if $\bar\Gamma$ is
empty.)}

\medskip
As a consequence of Corollary~5.4 we can calculate the expectation of
products of Wick polynomials of Gaussian random variables.

Let $X_{k,j}$, $EX_{k,j}=0$, $1\le k\le p$, $1\le j\le n_k$, 
be a sequence of (jointly) Gaussian random variables. We want 
to calculate the expected value of the product of the Wick 
polynomials $\colon\!X_{k,1}\cdots X_{k,n_k}\!\colon$,
\ $1\le k\le p$, if we know all covariances
$EX_{k,j}X_{\bar k,\bar j}=a((k,j),(\bar k,\bar j))$,  %\jmath
$1\le k,\bar k,\le p$, $1\le j\le n_k$, $1\le \bar j\le\bar n_k$. %\jmath
For this goal let us consider the class of closed diagrams
$\bar\Gamma(k_1,\dots,k_p)$, and define the following quantity
$\gamma(A)$ depending on the closed diagrams $\gamma$ and the set
$A$ of all covariances $EX_{k,j}X_{\bar k,\bar j}=a((k,j),    %\jmath
(\bar k,\bar j))$                                             %\jmath.
$$
\gamma(A)=\prod_{((k,j),(\bar k,\bar j))\textrm{ is an edge of }\gamma} %jmath
a((k,j),(\bar k,\bar j)), \quad \gamma\in\Gamma. %\jmath
$$
With the above notation we can formulate the following result.

\medskip\noindent
{\bf Corollary 5.5.} {\it
Let $X_{k,j}$, $EX_{k,j}=0$, $1\le k\le p$, $1\le j\le n_k$, be a
sequence of Gaussian random variables. Let
$a((k,j),(\bar k,\bar j))=EX_{k,j}X_{\bar k,\bar j}$,  %\jmath
$1\le k,\bar k,\le p$, $1\le j\le n_k$, $1\le \bar j\le\bar n_k$ %\jmath
denote the covariances of these random variables. Then
the expected value of the product of the Wick polynomials
$\colon\!X_{k,1}\cdots X_{k,n_k}\!\colon$, $1\le k\le p$, 
can be expressed as
$$
E\left(\prod_{k=1}^p\colon\!X_{k,1}\cdots X_{k,n_k}\!\colon\right)
=\sum_{\gamma\in\bar\Gamma(k_1,\dots,k_p)} \gamma(A)
$$
with the above defined quantities $\gamma(A)$. In the case when
$\bar\Gamma(k_1,\dots,k_p)$ is empty, e.g. if $k_1+\cdots+k_p$ is an
odd number, the above expectation equals zero.}

\medskip\noindent
{\it Remark.} In the special case when $X_{k,1}=\cdots=X_{k,n_k}=X_k$,
and $EX_k^2=1$ for all indices $1\le k\le p$ Corollary~5.5 provides
a formula for the expectation of the product of Hermite
polynomials of standard normal random variables. In this
case we have $a((k,j),(\bar k,\bar j))=\bar a(k,\bar k)$ with a %\jmath
function $\bar a(\cdot,\cdot)$ not depending on the arguments~$j$
and~$\bar j$, and the left-hand side of the identity in Corollary~5.5 %\jmath
equals $EH_{n_1}(X_1)\cdots H_{n_p}(X_p)$ with standard normal random
variables $X_1,\dots,X_n$ with correlations
$EX_kX_{\bar k}=\bar a(k,\bar k)$.

\medskip\noindent
{\it Proof of Corollary 5.5.}\/ We can represent the random variables
$X_{k,j}$ in the form $X_{k,j}=\sum\limits_pc_{k,j,p}\xi_p$ with some
appropriate coefficients $c_{k,j,p}$, where $\xi_1,\xi_2,\dots$ is
a sequence of independent standard normal random variables. Let
$Z(\,dx)$ denote a random spectral measure corresponding to the
one-dimensional spectral measure with density function
$g(x)=\frac1{2\pi}$ for $|x|<\pi$, and $g(x)=0$ for $|x|\ge\pi$. The
random integrals $\int e^{ipx}Z(\,dx)$, $p=0,\pm1,\pm2,\dots$, are
independent standard normal random variables.
Define $h_{k,j}(x)=\sum\limits_p c_{k,j,p}e^{ipx}$, $k=1,\dots,p$,
$1\le j\le n_k$. The random variables $X_{k,j}$ can be identified
with the random integrals $\int h_{k,j}(x)Z(\,dx)$, $k=1,\dots,p$,
$1\le j\le n_k$, since their joint distributions coincide. Put
$\hat h_k(x_1,\dots,x_{n_k})=\prod\limits_{j=1}^{n_k}h_{k,j}(x_j)$.
It follows from Theorem~4.7 that
$$
\colon\!X_{k,1}\cdots X_{k,n_k}\!\colon=
\int\hat h_k(x_1,\dots,x_{n_k}) Z(\,dx_1)\dots Z(\,dx_{n_k})
=n_k!I(\hat h_k(x_1,\dots,x_{n_k}))
$$
for all $1\le k\le p$. Hence an application of Corollary~5.4
yields Corollary~5.5. One only has to observe that
$\int_{-\pi}^\pi h_{k,j}(x)\overline{h_{\bar k,\bar j}(x)}\,dx %\jmath
=a((k,j),(\bar k,\bar j))$ for all $k,\,k=1,\dots,p$ and %\jmath
$1\le j\le n_k$. \hfill$\qed$

\medskip
Theorem~5.3 states in particular that the product of Wiener--It\^o
integrals with respect to a random spectral measure  of a stationary
Gaussian fields belongs to the Hilbert space ${\cal H}$ defined by this
field, since it can be written as a sum of Wiener--It\^o integrals.
This means a trivial measurability condition, and also that the
product has a finite second moment, which is not so trivial.
Theorem~5.3 actually gives the following non-trivial inequality.

Let $h_1\in{\cal H}_G^{n_1}$,\dots, $h_m\in{\cal H}_G^{n_m}$. Let
$|\bar\Gamma(n_1,n_1,\dots,n_m,n_m)|$ denote the number of complete
diagrams in $\bar\Gamma(n_1,n_1,\dots,n_m,n_m)$, and put
$$
C(n_1,\dots,n_m)=\frac{|\bar\Gamma(n_1,n_1,\dots,n_m,n_m)|}
{n_1!\cdots n_m!}.
$$
In the special case $n_1=\cdots=n_m=n$ let
$\bar C(n,m)=C(n_1,\dots,n_m)$. Then

\medskip\noindent
{\bf Corollary 5.6.} {\it
\begin{eqnarray*}
&&E\left[(n_1!I_G(h_1))^2\cdots (n_m! I_G(h_m))^2\right] \\
&&\qquad \le C(n_1,\dots,n_m) E(n_1!I_G(h_1))^2
\cdots (n_m! E(I_G(h_m))^2.
\end{eqnarray*}
In particular,
$$
E\left[(n!I_G(h))^{2m}\right]\le\bar C(n,m)(E(n!I_G(h))^2)^m
\quad \textrm{ if \ } h\in {\cal H}_G^n.
$$
}

\medskip\noindent
Corollary~5.6 follows immediately from Corollary~5.4 by applying it
first for the sequence $h_1,h_1,\dots,h_m,h_m$ and then for the pair
$h_j,h_j$ which yields that 
$$
E (n_j!I_G(h_j))^2=n_j!\|h_j\|^2, \quad 1\le j\le m.
$$ 
One only has to observe that
$|h_\gamma|\le\|h_1\|^2\cdots\|h_m\|^2$ for all complete diagrams
by Part~(A) of Theorem~5.3.

The inequality in Corollary~5.6 is sharp. If $G$ is a finite
measure and $h_1\in H_G^{n_1}$,\dots, $h_m\in H^{n_m}_G$ are constant
functions, then equality can be written in Corollary~5.6. We remark
that in this case $I_G(h_1),\dots,I_G(h_m)$ are constant times the
$n_1$-th,\dots, $n_m$-th Hermite polynomials of the same standard
normal random variable. Let us emphasize that the constant
$C(n_1,\dots,n_m)$ depends only on the parameters $n_1,\dots,n_m$
and not on the form of the functions $h_1,\dots,h_m$. The function
$C(n_1,\dots,n_m)$ is monotone in its arguments. The following
argument shows that
$$
{C(n_1+1,n_2,\dots,n_m)}\ge{C(n_1,\dots,n_m)}
$$

Let us call two complete diagrams in
$\bar\Gamma(n_1,n_1,\dots,n_m,n_m)$ or in
$\bar\Gamma(n_1+1,n_1+1,\dots,n_m,n_m)$ equivalent if they
can be transformed into each other by permuting the vertices
$(1,1),\dots,(1,n_1)$ in $\bar\Gamma(n_1,n_1,\dots,n_m,n_m)$ or
the vertices $(1,1),\dots,(1,n_1+1)$ in
$\bar\Gamma(n_1+1,n_1+1,\dots,n_m,n_m)$. The equivalence classes
have $n_1!$ elements in the first case and $(n_1+1)!$ elements
in the second one. Moreover, the number of equivalence classes is
less in the first case than in the second one. (They would agree if
we counted only those equivalence classes in the second case which
contain a diagram where $(1,n_1+1)$ and $(2,n_1,1)$ are connected
by an edge. Hence
$$
\frac1{n_1!}|\bar\Gamma(n_1,n_1,\dots,n_m,n_m)|\le
\frac1{(n_1+1)!}|\bar\Gamma(n_1+1,n_1+1,\dots,n_m,n_m)|
$$
as we claimed.

The next result may better illuminate the content of Corollary~5.6.

\medskip\noindent
{\bf Corollary 5.7.} {\it Let $\xi_1,\dots,\xi_k$ be a normal random
vector, and $P(x_1,\dots,x_k)$ a polynomial of degree~$n$. Then
$$
E\left[P(\xi_1,\dots,\xi_k)^{2m}\right]\le\bar C(n,m)(n+1)^m
\left(EP(\xi_1,\dots,\xi_k)^2\right)^m
$$
with the constant $\bar C(n,m)$ introduced before Corollary~5.6.}

\medskip
The multiplying constant $\bar C(n,m)(n+1)^m$ is not sharp in this
case. 

\medskip\noindent
{\it Proof of Corollary 5.7.}\/ We can write $\xi_j=\int f_j(x)Z(\,dx)$
with some $f_j\in{\cal H}^1$, $j=1,2,\dots,k$, where $Z(\,dx)$ is
the same as in the proof of Corollary~5.5. There exist some
$h_j\in{\cal H}^j$, $j=0,1,\dots,n$, such that
$$
P(\xi_1,\dots,\xi_k)=\sum_{j=0}^n j! I(h_j).
$$
Then
\begin{eqnarray*}
&&EP(\xi_1,\dots,\xi_k)^{2m}
=E\left[\left(\sum_{j=0}^nj!I(h_j)\right)^{2m}\right]
\le (n+1)^m E\left[\sum_{j=0}^n (j!I(h_j))^2\right]^m   \\
&&\qquad\le (n+1)^m\sum_{p_1+\cdots+p_n=m}C(p_1,\dots,p_n)(EI(h_0)^2)^{p_0}
\cdots (En!I(h_n)^2)^{p_n}\frac{m!}{p_1!\cdots p_n!} \\
&&\qquad\le (n+1)^m\bar C(n,m)\sum_{p_1+\cdots+p_n=m} (EI(h_0)^2)^{p_0}
\cdots (EI(n!h_n)^2)^{p_n}\frac{m!}{p_1!\cdots p_n!} \\
&&\qquad=(n+1)^m\bar C(n,m)
\left[\sum E (j!I(h_j))^2\right]^m=(n+1)^m \bar C(n,m)
\left(EP(\xi_1,\dots,\xi_k)^2\right)^m.
\end{eqnarray*}
\hfill$\qed$

\chapter{Subordinated Random Fields: Construction of
Self-similar Fields}

Let $X_n$, $n\in\textrm{\BBB Z}_\nu$, be a discrete stationary 
Gaussian random field with a non-atomic spectral measure, and 
let the random field $\xi_n$, $n\in\textrm{\BBB Z}_\nu$, be 
subordinated to it. Let $Z_G$ denote the random spectral 
measure adapted to the random field $X_n$. By Theorem~4.2 
the random variable $\xi_0$ can be represented as
$$
\xi_0=f_0+\sum_{k=1}^\infty \frac1{k!}\int
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k)
$$
with an appropriate $f=(f_0,f_1,\dots)\in\,\textrm{Exp}\,{\cal H}_G$
in a unique way. This formula together with Theorem~4.4 yields the
following

\medskip\noindent
{\bf Theorem 6.1.} {\it A random field $\xi_n$, $n\in\textrm{\BBB Z}_\nu$,
subordinated to the stationary Gaussian random field $X_n$,
$n\in\textrm{\BBB Z}_\nu$, with non-atomic spectral measure can be 
written in the form
\begin{equation}
\xi_n=f_0+\sum_{k=1}^\infty \frac1{k!}\int e^{i((n,x_1+\cdots+x_k)}
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k), 
\quad n\in{\BBB Z}_\nu, \label{(6.1)}
\end{equation}
with some $f=(f_0,f_1,\dots)\in\,\textrm{\rm Exp}\,{\cal H}_G$,
where $Z_G$ is the random spectral measure adapted to the random 
field~$X_n$. This representation is unique. It is also clear 
that formula~(\ref{(6.1)}) defines a subordinated field for all 
$f\in\,\textrm{\rm Exp}\,{\cal H}_G$.}
\index{random field subordinated to a stationary Gaussian 
random field (discrete case)} 

\medskip\noindent
Let $G$ denote the spectral measure of the underlying stationary
Gaussian random field. If it has the property $G(\{x\colon\;x_p=u\})=0$
for all $u\in R^1$ and $1\le p\le\nu$, where $x=(x_1,\dots,x_\nu)$
(this is a strengthened form of the non-atomic property of $G$), 
then the functions
$$
\bar f_k(x_1,\dots,x_k)=f_k(x_1,\dots,x_k)
\tilde\chi_0^{-1}(x_1+\cdots+x_k),
\quad k=1,2,\dots,
$$
are meaningful, as functions in the measure space
$(R^{k\nu},{\cal B}^{k\nu},G^k)$, where
$\tilde\chi_n(x)=e^{i(n,x)}
\prod\limits_{p=0}^\nu\frac{e^{ix^{(p)}}-1}{ix^{(p)}}$,
$n\in\textrm{\BBB Z}_\nu$, denotes the Fourier transform of 
the indicator function of the $\nu$-dimensional unit cube
$\prod\limits_{p=1}^\nu[n^{(p)},n^{(p)}+1]$. Then the random
variable $\xi_n$ in formula~(\ref{(6.1)}) can be rewritten 
in the form
\begin{equation}
\xi_n=f_0+\sum_{k=1}^\infty \frac1{k!}\int\tilde\chi_n(x_1+\cdots+x_k)
\bar f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k),\quad n\in{\BBB Z}_\nu.
\label{(6.1a)}
\end{equation}
Hence the following Theorem~$6.1'$ can be considered as the continuous
time version of Theorem~6.1.

\medskip\noindent
{\bf Theorem~$6.1'$.} {\it Let a generalized random field 
$\xi(\varphi)$, $\varphi\in{\cal S}$, be subordinated to a 
stationary Gaussian generalized random field $X(\varphi)$, 
$\varphi\in{\cal S}$. Let $G$ denote the spectral measure 
of the field $X(\varphi)$, and let $Z_G$ be the random 
spectral measure adapted to it. Let the spectral measure~$G$
be non-atomic. Then $\xi(\varphi)$ can be written in the form
\begin{equation}
\xi(\varphi)=f_0\cdot\tilde\varphi(0)+\sum_{k=1}^\infty\frac1{k!}
\int\tilde\varphi(x_1+\cdots+x_k)f_k(x_1,\dots,x_k)Z_G(\,dx_1)
\dots Z_G(\,dx_k), \label{($6.1'$)}
\end{equation}
where the functions $f_k$ are invariant under all permutations 
of their variables,
$$
f_k(-x_1,\dots,-x_k)=\overline{f_k(x_1,\dots,x_k)}, 
\quad k=1,2,\dots,
$$
and
\begin{equation}
\sum_{k=1}^\infty \frac1{k!}\int (1+|x_1+\cdots+x_k|^2)^{-p}
|f_k(x_1+\cdots+x_k)|^2G(\,dx_1)\dots G(\,dx_k)<\infty \label{(6.2)}
\end{equation}
with an appropriate number $p>0$. This representation is unique.

Contrariwise, all random fields $\xi(\varphi)$, $\varphi\in{\cal S}$,
defined by formulas~(\ref{($6.1'$)}) and~(\ref{(6.2)}) 
are subordinated to the stationary, Gaussian random field 
$X(\varphi)$,~$\varphi\in{\cal S}$.}
\index{random field (generalized) subordinated to a generalized 
stationary Gaussian random field} 

\medskip\noindent
{\it Proof of Theorem~6.1$'$}.\/ The proof is based on the 
same ideas as the proof of Theorem~6.1. But here we also 
adapt some  arguments from the theory of generalized 
functions (see \cite{r15}). In particular, we exploit 
the following continuity property of generalized random 
fields and subordinated generalized random fields. If
$\varphi_n\to\varphi$ in the topology of the Schwartz 
space ${\cal S}$, and $X(\varphi)$, $\varphi\in{\cal S}$, 
is a generalized random field, then 
$X(\varphi_n)\Rightarrow X(\varphi)$
stochastically. If $X(\varphi)$, $\varphi\in{\cal S}$, is
a generalized Gaussian random field, then also the relation
$E[X(\varphi_n)-X(\varphi)]^2\to0$ holds in this case.
Similarly, if $\xi(\varphi)$, $\varphi\in{\cal S}$, is
a subordinated generalized random field, and 
$\varphi_n\to\varphi$, then 
$E[\xi(\varphi_n)-\xi(\varphi)]^2\to0$ by the definition 
of subordinated fields.

It can be seen with some work that a random field 
$\xi(\varphi)$,~$\varphi\in{\cal S}$, defined 
by~(\ref{($6.1'$)}) and~(\ref{(6.2)}) is subordinated 
to~$X(\varphi)$. One has to check that the definition 
of~$\xi(\varphi)$ in formula~(\ref{($6.1'$)}) is meaningful 
for all~$\varphi\in{\cal S}$, because of~(\ref{(6.2)}),
$\xi(T_t\varphi)=T_t\xi(\varphi)$ for all shifts~$T_t$,~$t\in R^\nu$,
by Theorem~4.4, and also the following continuity property holds. 
For all $\varepsilon>0$ there is a small neighbourhood~$H$ 
of the origin in the space~${\cal S}$ such that if 
$\varphi=\varphi_1-\varphi_2\in H$ for
some $\varphi_1,\varphi_2\in{\cal S}$ then
$E[\xi(\varphi_1)-\xi(\varphi_2)]^2=E\xi(\varphi)^2<\varepsilon^2$.

Since the Fourier transform 
$\varphi(\cdot)\to\tilde\varphi(\cdot)$ is a bicontinuous 
map in ${\cal S}$, to prove the above continuity property 
it is enough to check that $E\xi(\varphi)^2<\varepsilon^2$ 
if $\tilde\varphi\in H$ for an appropriate small 
neighbourhood~$H$ of the origin in~${\cal S}$. But this 
relation holds with the choice
$H=\{\varphi\colon (1+|x|^2)^p|\varphi(x)|\le\frac{\varepsilon^2}K
\textrm{ for all }x\in R^\nu\}$ with a sufficiently 
large $K>0$ because of condition~(\ref{(6.2)}).

To prove that all subordinated fields have the above 
representation observe that the relation
\begin{equation}
\xi(\varphi)=\Psi_{\varphi,0}+\sum_{k=1}^\infty\frac1{k!}\int
\Psi_{\varphi,k}(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k) 
\label{(6.3)}
\end{equation}
holds for all $\varphi\in{\cal S}$ with some
$(\Psi_{\varphi,0},\Psi_{\varphi,1},\dots)\in\,\textrm{Exp}\,{\cal H}_G$
depending on the function~$\varphi$. We are going to show 
that these functions $\Psi_{\varphi,k}$ can be given in the form
$$
\Psi_{\varphi,k}(x_1,\dots,x_k)=f_k(x_1,\dots,x_k)
\cdot\tilde\varphi (x_1+\cdots+x_k), \quad k=1,2,\dots,
$$
with some functions $f_k\in{\cal B}^{k\nu}$, and
$$
\Psi_{\varphi,0}=f_0\cdot\tilde\varphi(0)
$$
for all $\varphi\in{\cal S}$ with a sequence of functions
$f_0,f_1,\dots$ not depending on~$\varphi$.

To show this let us choose a $\varphi_0\in{\cal S}$  such 
that $\tilde\varphi_0(x)>0$ for all $x\in R^{\nu}$. (We 
can make for instance the choice $\varphi_0(x)=e^{-(x,x)}$.) 
We claim that the finite linear combinations
$\sum a_p\varphi_0(x-t_p)=\sum a_pT_{t_p}\varphi_0(x)$ are
dense in ${\cal S}$. To prove this it is enough to show 
that if the Fourier transform $\tilde\psi$ of a function 
$\psi\in{\cal S}$ has a compact support, then in every open 
neighbourhood of $\psi$ (in the topology of the space 
${\cal S}$) there is a function of the form
$\sum a_p\varphi_0(x-t_p)$. Indeed, this implies that the 
above introduced linear combinations constitute a dense 
subclass of ${\cal S}$, since the functions $\psi$ with 
the above property are dense in ${\cal S}$. (The statement 
that these functions $\psi$ are dense in ${\cal S}$ is 
equivalent to the statement that their Fourier transforms 
$\tilde\psi$ are dense in the space 
$\tilde{{\cal S}}\subset{\cal S}^c$ consisting of 
the Fourier transforms of the (real valued) functions in 
the space~${\cal S}$.) We have
$\frac{\tilde\psi}{\tilde\varphi_0}\in{\cal S}^c$ for such
functions~$\psi$, where ${\cal S}^c$ denotes the 
Schwartz-space of complex valued, at infinity strongly 
decreasing, smooth functions again, because 
$\tilde\varphi_0(x)\neq0$, and $\tilde\psi$ has a compact 
support. There exists a function $\chi\in{\cal S}$ such 
that $\tilde\chi=\frac{\tilde\psi}{\tilde\varphi_0}$. 
(Here we exploit that the space of Fourier transforms of 
the functions from ${\cal S}$ agrees with the space of 
those functions $f\in{\cal S}^c$ for which
$f(-x)=\overline{f(x)}$.) Therefore
$\psi(x)=\chi*\varphi_0(x)=\int \chi(t)\varphi_0(x-t)\,dt$, 
where $*$ denotes convolution. It can be seen by 
exploiting this relation together with the rapid 
decrease of~$\chi$ and~$\varphi_0$ together of its 
derivatives at infinity, and approximating the 
integral defining the convolution by an appropriate 
finite sum that for all integers $r>0$, $s>0$ and 
real numbers~$\varepsilon>0$ there exists a finite 
linear combination 
$\hat\psi(x)=\hat\psi_{r,s,\varepsilon}(x)
=\sum\limits_p a_p\varphi_0(x-t_p)$ such
that $(1+|x|^s)|\psi(x)-\hat\psi(x)|<\varepsilon$ 
for all $x\in R^\nu$, and the same estimate holds for 
all derivatives of $\psi(x)-\hat\psi(x)$ of order 
less than~$r$. 

I only briefly explain why such an approximation exists. 
Some calculation enables us to reduce this statement to 
the case when $\psi=\chi*\varphi_0$ with a function 
$\chi\in{\cal D}$, which has compact support. To give 
the desired approximation choose a small number 
$\delta>0$, introduce the cube
$\Delta=\Delta(\delta)=[-\delta,\delta)^\nu\subset R^\nu$ 
and define the vectors 
$k(\delta)=(2k_1\delta,\dots,2k_\nu\delta)\in R^\nu$
for all $k=(k_1,\dots,k_\nu)\in\textrm{\BBB Z}_\nu$.
Given a fixed vector $x\in R^\nu$ let us define the vector
$u(x)\in R^\nu$ for all $u\in R^\nu$ as $u(x)=x+k(\delta)$ 
with that vector $k\in\textrm{\BBB Z}_\nu$ for which 
$x+k(\delta)-u\in\Delta$, and put 
$\varphi_{0,x}(u)=\varphi_0(u(x))$. It can be seen that
$\hat\psi(x)=\chi*\varphi_{0,x}(x)$ is a finite linear 
combination of numbers of the form $\varphi_0(x-t_k)$ 
(with $t_k=k(\delta)$) with coefficients not depending 
on~$x$. Moreover, if $\delta>0$ is chosen sufficiently 
small (depending on $r,s$  and~$\varepsilon$), then 
$\hat\psi(x)=\hat\psi_{r,s,\varepsilon}(x)$ has all 
properties we demanded.

The above argument implies that there is a sequence of 
functions $\hat\psi_{r,s,\varepsilon}$ which converges to 
the function~$\psi$ in the topology of the space~${\cal S}$. 
As a consequence, the finite linear combinations 
$\sum a_p\varphi_0(x-t_p)$ are dense in~${\cal S}$.

Define
$$
f_k(x_1,\dots,x_k)
=\frac{\Psi_{\varphi_0,k}(x_1,\dots,x_k)}
{\tilde\varphi_0(x_1+\cdots+x_k)},
\quad k=1,2,\dots,\quad \textrm{and }
f_0=\frac{\Psi_{\varphi_0,0}}{\tilde\varphi_0(0)}.
$$
If $\varphi(x)=\sum a_p\varphi_0(x-t_p)=\sum a_pT_{t_p}\varphi_0(x)$,
and the sum defining~$\varphi$ is finite, then by Theorem~4.4
\begin{eqnarray*}
\xi(\varphi)&&=\left(\sum a_p\right)f_0\cdot\tilde\varphi_0(0)
+\sum_{k=1}^\infty\frac1{k!}\int
\sum_p a_pe^{i(t_p,x_1+\cdots+x_k)}
\tilde\varphi_0(x_1+\cdots+x_k) \\
&&\qquad\qquad\qquad\qquad\qquad\cdot f_k(x_1,\dots,x_k)
Z_G(\,dx_1)\dots Z_G(\,dx_k)\\
&&=f_0\cdot\tilde\varphi(0)+\sum_{k=1}^\infty\frac1{k!}\int
\tilde\varphi(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k).
\end{eqnarray*}
Relation~(\ref{(6.3)}) holds for all $\varphi\in{\cal S}$, 
and there exists a sequence of functions
$\varphi_j(x)=\sum\limits_p a_p^{(j)}\varphi_0(x-t_p^{(j)})\in{\cal S}$
satisfying~(\ref{($6.1'$)}) such that $\varphi_j\to\varphi$ 
in the topology of~${\cal S}$. This implies that
$\lim E[\xi(\varphi_j)-\xi(\varphi)]^2\to0$, and in particular
$EI_G(\Psi_{\varphi,k}-\hat\varphi_{j,k}f_k)^2\to0$ with
$\hat\varphi_{j,k}(x_1,\dots,x_k)=\tilde\varphi_j(x_1+\cdots+x_k)$
as $j\to\infty$ for all $k=1,2,\dots$. In the subsequent steps
of the proof we restrict the domain of integration to bounded 
sets~$A$, because this enables us to carry out some limiting 
procedures needed in our argument. We can write that
$$
\int_A|\Psi_{\varphi,k}(x_1,\dots,x_k)-\tilde\varphi_j(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)|^2G(\,dx_1)\dots G(\,dx_k)\to0
$$
as $j\to\infty$ for all $k$ and for all bounded sets 
$A\in R^{k\nu}$. On the other hand,
$$
\int_A|\tilde\varphi(x_1+\cdots+x_k)-\tilde\varphi_j(x_1+\cdots+x_k)|^2
|f_k(x_1,\cdots,x_k)|^2G(\,dx_1)\dots G(\,dx_k)\to0,
$$
since $\tilde\varphi_j(x)-\tilde\varphi(x)\to0$ in the supremum
norm if $\tilde\varphi_j\to\tilde\varphi$ in the topology 
of ${\cal S}$, and the property $\tilde\varphi_0(x)>0$ (of the
function $\tilde\varphi_0$ appearing in the definition of the
function $f_k$)  together with the continuity of 
$\tilde\varphi_0$ and the inequality 
$EI_G(\hat\varphi_{0,k}f_k)^2<\infty$ imply that
$\int_A |f_k(x_1,\dots,x_k)|^2G(\,dx_1)\dots G(dx_k)<\infty$
on all bounded sets~$A$. The last two relations yield that
$$
\Psi_{\varphi,k}(x_1,\dots,x_k)
=\tilde\varphi(x_1+\cdots+x_k)f_k(x_1,\dots,x_k), 
\quad k=1,2,\dots,
$$
since both sides of this identity is the limit of the sequence
$$
\tilde\varphi_j(x_1+\cdots+x_k)f_k(x_1,\dots,x_k), \quad j=1,2,\dots
$$ 
in the $L^2_{G^k_A}$ norm, where $G^k_A$ denotes the restriction of
the measure~$G^k$ to the set~$A$.

Similarly,
$$
\psi_{\varphi,0}=\tilde\varphi(0)f_0.
$$
These relations imply~(\ref{($6.1'$)}).

To complete the proof of Theorem~$6.1'$ we show that~(\ref{(6.2)}) 
follows from the continuity of the transformation
$F\colon\;\varphi\to \xi(\varphi)$ from the space 
${\cal S}$ into the space $L_2(\Omega,{\cal A},P)$.

We recall that the transformation $\varphi\to\tilde\varphi$ is
bicontinuous in ${\cal S}^c$. Hence for a subordinated field
$\xi(\varphi)$, $\varphi\in{\cal S}$, the transformation
$\tilde\varphi\to\xi(\varphi)$ is a continuous map from the 
space of the Fourier transforms of the functions in the 
space~${\cal S}$ to $L_2(\Omega,{\cal A},P)$. This continuity 
implies that there exist some integers $p>0$, $r>0$ and real 
number $\delta>0$ such that if
\begin{equation}
(1+|x^2|)^p\left|\frac{\partial^{s_1+\cdots+s_\nu}}
{\partial {x^{(1)}}^{s_1}
\dots\partial {x^{(\nu)}}^{s_\nu}}\tilde\varphi(x)\right|<\delta
\quad\textrm{for all } s_1+\cdots+s_\nu\le r, \label{(6.4)}
\end{equation}
then $E\xi(\varphi)^2\le1$.

Let us choose a function $\psi\in{\cal S}$ such that $\psi$ 
has a compact support, $\psi(x)=\psi(-x)$, $\psi(x)\ge0$ 
for all $x\in R^\nu$, and $\psi(x)=1$ if $|x|\le 1$. (There 
exist such functions.) Define the functions 
$\tilde\varphi_m(x)=C(1+|x|^2)^{-p}\psi(\frac xm)$.
Then $\varphi_m\in{\cal S}$, since its Fourier transform
$\tilde\varphi_m$ is an even function, and it is in the
space~${\cal S}$ being an infinite many times 
differentiable function with compact support. Moreover, 
$\varphi_m$ satisfies~(\ref{(6.4)}) for all $m=1,2,\dots$ 
if the number $C>0$ in its definition is chosen 
sufficiently small. This number~$C$ can be chosen
independently of~$m$. (To see this observe that 
$(1+|x^2|)^{-p}$ together with all of its derivatives 
of order not bigger than~$r$ can be bounded by 
$\frac{C(p,r)}{(1+|x|^2)^p}$ with an appropriate
constant~$C(p,r)$.) Hence
$$
E\xi(\varphi_m)^2=\sum\frac1{k!}
\int|\tilde\varphi_m(x_1+\cdots+x_k)|^2
|f_k(x_1,\cdots,x_k)|^2G(dx_1)\dots G(dx_k)\le1
$$
for all $m=1,2,\dots$.

As $\tilde\varphi_m(x)\to C(|1+|x|^2)^{-p}$ as $m\to\infty$, 
and $\tilde\varphi_k(x)\ge0$, an $m\to\infty$ limiting 
procedure in the last relation together with Fatou's 
lemma imply that
$$
C\sum\frac1{k!}\int(1+|x_1+\cdots+x_k)|^2)^{-p}
|f_k(x_1,\cdots,x_k)|^2G(dx_1)\dots G(dx_k)\le1.
$$
Theorem~$6.1'$ is proved. \hfill$\qed$

\medskip
We shall call the representations given in Theorems~6.1 
and~$6.1'$ the canonical representation of a subordinated field. 
\index{canonical representation of a subordinated field}
From now on we restrict ourselves to the case $E\xi_n=0$ or 
$E\xi(\varphi)=0$ respectively, i.e. to the case when $f_0=0$ 
in the canonical representation. If
$$
\xi(\varphi)=\sum_{k=1}^\infty\frac1{k!}\int\tilde\varphi(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k),
$$
then
$$
\xi(\varphi^A_t)=\sum_{k=1}^\infty\frac1{k!}\frac{t^\nu}{A(t)}
\int\tilde\varphi(t(x_1+\cdots+x_k))
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k)
$$
with the function $\varphi^A_t$ defined in~(\ref{(1.3)}). Define 
the spectral measures $G_t$ by the formula $G_t(A)=G(tA)$. Then 
we have by Lemma~4.6
$$
\xi(\varphi^A_t)\stackrel{\Delta}{=}\sum_{k=1}^\infty\frac1{k!}
\frac{t^\nu}{A(t)}
\int\tilde\varphi(x_1+\cdots+x_k)
f_k\left(\frac{x_1}t,\dots,\frac{x_k}t\right)
Z_{G_t}(\,dx_1)\dots Z_{G_t}(\,dx_k).
$$

If $G(tB)=t^{2\kappa}G(B)$ with some $\kappa>0$ for all $t>0$ and
$B\in{\cal B}^\nu$, $f_k(\lambda x_1,\dots,\lambda x_k)=
\lambda^{\nu-\kappa k-\alpha}f_k(x_1,\dots,x_k)$, and $A(t)$ is
chosen as $A(t)=t^\alpha$, then Theorem~4.5 (with the choice
$G'(B)=G(tB)=t^{2\kappa}G(B)$) implies that
$\xi(\varphi_t^A)\stackrel{\Delta}{=}\xi(\varphi)$.
Hence we obtain the following

\medskip\noindent
{\bf Theorem 6.2.} {\it Let a generalized random 
field $\xi(\varphi)$ be given by the formula
\begin{equation}
\xi(\varphi)=\sum_{k=1}^\infty\frac1{k!}\int \tilde\varphi
(x_1+\cdots+x_k)f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k).
\label{(6.5)}
\end{equation}
If $f_k(\lambda x_1,\dots,\lambda x_k)=\lambda^{\nu-\kappa k-\alpha}
f_k(x_1,\dots,x_k)$ for all~$k$, $(x_1,\dots,x_k)\in R^{k\nu}$ and
$\lambda>0$, $G(\lambda A)=\lambda^{2\kappa}G(A)$ for all
$\lambda>0$ and $A\in{\cal B}^\nu$, then $\xi$ is a self-similar
random field with parameter $\alpha$.}

\medskip
The discrete time version of this result can be proved in the same
way. It states the following

\medskip\noindent
{\bf Theorem 6.2$'$.} {\it If a discrete random field $\xi_n$,
$n\in\textrm{\BBB Z}_\nu$, has the form
\begin{equation}
\xi_n=\sum_{k=1}^\infty \frac1{k!}\int\tilde\chi_n(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k), \quad
n\in\textrm{\BBB Z}_\nu, \label{($6.5'$)}
\end{equation}
and $f_k(\lambda x_1,\dots,\lambda x_k)=\lambda^{\nu-\kappa k-\alpha}
f_k(x_1,\dots,x_k)$ for all~$k$, $G(\lambda A)=\lambda^{2\kappa}G(A)$,
then $\xi_n$ is a self-similar random field with parameter~$\alpha$.}

\medskip
Theorems~6.2 and~$6.2'$ enable us to construct self-similar random
fields. Nevertheless, we have to check whether formulas~(\ref{(6.5)})
and~(\ref{($6.5'$)})  are meaningful. The hard part of this problem is
to check whether
$$
\sum \frac1{k!}\int |\tilde\chi_n(x_1+\cdots+x_k)|^2
|f_k(x_1,\dots,x_k)|^2 G(\,dx_1)\dots G(\,dx_k)<\infty
$$
in the discrete time case or whether
$$
\sum \frac1{k!}\int |\tilde\varphi(x_1+\cdots+x_k)|^2
|f_k(x_1,\dots,x_k)|^2 G(\,dx_1)\dots G(\,dx_k)<\infty
\quad \textrm{for all }\varphi\in{\cal S}
$$
in the generalized field case.

It is rather hard to decide in the general case when these 
expressions are finite. The next result enables us to prove
the finiteness of these expressions in some interesting cases.

Let us define the measure~$G$
\begin{equation}
G(A)=\int_A |x|^{2\kappa-\nu}a\left(\frac x{|x|}\right)\,dx,
\quad A\in{\cal B}^\nu,
\label{(6.6)}
\end{equation}
where $a(\cdot)$ is a non-negative, measurable and even 
function on the $\nu$-dimensional unit sphere $S_{\nu-1}$, 
and $\kappa>0$. (The condition $\kappa>0$ is imposed to 
guarantee the relation $G(A)<\infty$ for all bounded sets 
$A\in{\cal B}^\nu$.) We prove the following

\medskip\noindent
{\bf Proposition 6.3.} {\it Let the measure $G$ be defined 
in formula~(\ref{(6.6)}).

\medskip
\begin{description}
\item[(a)] If the function $a(\cdot)$ is bounded on the 
unit sphere $S_{\nu-1}$, and $\frac \nu k>2\kappa>0$, then
$$
D(n)=\int|\tilde\chi_n(x_1+\cdots+x_k)|^2 
G(\,dx_1)\dots G(\,dx_k)<\infty
\quad \textrm{for all \ } n\in\textrm{\BBB Z}_\nu,
$$
and
\begin{eqnarray*}
D(\varphi)&&=\int|\tilde\varphi(x_1+\cdots+x_k)|^2
G(\,dx_1)\dots G(\,dx_k)\\
&&\le C\int(1+|x_1+\cdots+x_k)|^2)^{-p} 
G(\,dx_1)\dots G(\,dx_k)<\infty
\end{eqnarray*}
for all $\varphi\in{\cal S}$ and $p>\frac\nu2$ with some
$C=C(\varphi,p)<\infty$.
\item[(b)] If there is a constant $C>0$ such that $a(x)>C$ in a
neighbourhood of a point $x_0\in S_{\nu-1}$, and either 
$2\kappa\le0$ or $2\kappa\ge\frac\nu k$, then the integrals $D(n)$ 
are divergent, and the same relation holds for $D(\varphi)$ 
with some $\varphi\in{\cal S}$.
\end{description}

}

\medskip\noindent
{\it Proof of Proposition 6.3.}\/ {\it Proof of Part (a).}

\noindent
We may assume that $a(x)=1$ for all $x\in S_{\nu-1}$. Define
$$
J_{\kappa,k}(x)=\int_{x_1+\cdots+x_k=x}
|x_1|^{2\kappa-\nu}\cdots|x_k|^{2\kappa-\nu}
\,dx_1\dots\,dx_k, \quad x\in R^{\nu},
$$
for $k\ge2$, where $\,dx_1\dots\,dx_k$ denotes the Lebesgue
measure on the hyperplane $x_1+\cdots+x_k=x$, and let
$J_{\kappa,1}(x)=|x|^{2\kappa-\nu}$. We have
$$
J_{\kappa,k}(\lambda x)
=|\lambda|^{k(2\kappa-\nu)+(k-1)\nu}J_{\kappa,k}(x),
=|\lambda|^{2k\kappa-\nu}J_{\kappa,k}(x),
\quad x\in R^\nu\;\; \lambda>0,
$$
because of the homogeneity of the integral. We can write, because
of (\ref{(6.6)}) with $a(x)\equiv1$
\begin{equation}
D(n)=\int_{R^\nu}|\tilde\chi_n(x)|^2 J_{\kappa,k}(x)\,dx, 
\label{(6.7)}
\end{equation}
and
$$
D(\varphi)=\int_{R^\nu}|\tilde\varphi(x)|^2J_{\kappa,k}(x)\,dx.
$$
%&&\int (1+|x_1+\cdots+x_k|^2)^{-p}G(\,dx_1)\dots G(\,dx_k)
%=\int (1+|x|^2)^{-p} J_{\kappa,k}(x)\,dx. \nonumber 
%\end{eqnarray}
We prove by induction on $k$ that
\begin{equation}
J_{\kappa,k}(x)\le C(\kappa,k)|x|^{2\kappa k-\nu} \label{(6.8)}
\end{equation}
with an appropriate constant $C(\kappa,k)<\infty$ if
$\frac\nu k>2\kappa>0$.

Inequality~(\ref{(6.8)}) holds for $k=1$, and we have
$$
J_{\kappa,k}(x)=\int J_{\kappa,k-1}(y)|x-y|^{2\kappa-\nu}\,dy
$$
for $k\ge2$. Hence
\begin{eqnarray*}
J_{\kappa,k}(x)&\le& C(\kappa,k-1)\int
|y|^{(2\kappa(k-1)-\nu}|x-y|^{2\kappa-\nu}\,dy\\
&=& C(\kappa,k-1)|x|^{2\kappa k-\nu}
\int |y|^{(2\kappa(k-1)-\nu}\left|\frac x{|x|}-y\right|^{2\kappa-\nu}\,dy
=C(\kappa,k)|x|^{2\kappa k-\nu},
\end{eqnarray*}
since $\int |y|^{(2\kappa(k-1)-\nu}
\left|\frac x{|x|}-y\right|^{2\kappa-\nu}\,dy<\infty$.

The last integral is finite, since its integrand behaves at zero
asymptotically as $C|y|^{2\kappa(k-1)-\nu}$, at the point
$e=\frac x{|x|}\in S_{\nu-1}$ as $C_2|y-e|^{2\kappa-\nu}$ and at
infinity as $C_3|y|^{2\kappa k-2\nu}$. Relations~(\ref{(6.7)}) 
and~(\ref{(6.8)}) imply that
\begin{eqnarray*}
D(n)&\le& C'\int |\tilde\chi_0(x)|^2|x|^{2\kappa k-\nu}\,dx
\le C''\int |x|^{2\kappa k-\nu}\prod_{l=1}^\nu\frac1{1+|x^{(l)}|^2}\,dx\\
&\le& C'''\int_{|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|}
|x^{(1)}|^{2\kappa k-\nu}\prod_{l=1}^\nu\frac1{1+|x^{(l)}|^2}\,dx\\
&=&\sum_{p=0}^\infty C'''
\int_{|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|,
\;2^p\le |x^{(1)}|<2^{p+1}}
+C'''\int_{|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|, |x^{(1)}|<1}.
\end{eqnarray*}
The second term in the last sum can be simply bounded by a constant, 
since $B=\left\{x\colon\;|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|, \;
|x^{(1)}|<1\right\}\subset\{x\colon\;|x|\le\sqrt \nu\}$,
and $|x^{(1)}|^{2\kappa k-\nu}\prod\limits_{l=1}^\nu\frac1{1+|x^{(l)}|^2}
\le\textrm{const.}\, |x|^{2\kappa k-\nu}$ on the set~$B$. Hence
$$
D(n)\le C_1\sum_{p=0}^\infty 2^{p(2\kappa k-\nu)}
\left[\int_{-\infty}^\infty\frac 1{1+x^2}\,dx\right]^\nu+C_2<\infty.
$$
We have $|\varphi(x)|\le C(1+|x^2|)^{-p}$ with some $C>0$ and $D>0$
if $\varphi\in{\cal S}$.
The proof of the estimate $D(\varphi)<\infty$ for $\varphi\in{\cal S}$
is similar but simpler.

\medskip\noindent
{\it Proof of Part (b).}\/ Define, similarly to the function
$J_{\kappa,k}$ the function
$$
J_{\kappa,k,a}(x)=\int_{x_1+\cdots+x_k=x}
|x_1|^{2\kappa-\nu}a\left(\frac{x_1}{|x_1|}\right)
\cdots|x_k|^{2\kappa-\nu} a\left(\frac{x_k}{|x_k|}\right)
\,dx_1\dots\,dx_k, \quad x\in R^{\nu},
$$
for $k\ge2$, where $\,dx_1\dots\,dx_k$ denotes the Lebesgue 
measure on the hyperplane $x_1+\cdots+x_k=x$, and put
$J_{1,a}(x)=x^{2\kappa-\nu}a(\frac x{|x|})$. We can prove
by induction with respect to~$k$ that
\begin{eqnarray*}
J_{\kappa,k,a}(x)
&\ge&\int_{y\colon\; (\frac12-\alpha)|x|<|y|<(\frac12+\alpha)|x|}
J_{\kappa,k-1,a}(y) 
a\left(\frac{x-y}{|x-y|}\right)|x-y|^{2\kappa-\nu}\,dy \\
&\ge& C(\kappa,k,a(\cdot))
\int_{y\colon\; (\frac12-\alpha)|x|<|y|<(\frac12+\alpha)|x|}
|x|^{2\kappa(k-1)-\nu}|x-y|^{2\kappa-\nu}\,dy
\end{eqnarray*}
with the choice of some number $0<\alpha<\frac12$ and 
\begin{eqnarray*}
J_{\kappa,k,a}(x)&\ge&\int_{y\colon\; |y|>2x}
J_{\kappa,k-1,a}(y) a\left(\frac{x-y}{|x-y|}\right)
|x-y|^{2\kappa-\nu}\,dy \\
&\ge& C(\kappa,k,a(\cdot))\int_{y\colon\; |y|>2x}
|x-y|^{2\kappa k-2\nu}\,dy
\end{eqnarray*}
with some coefficient $C(\kappa,k,a(\cdot))$ if 
$\frac x{|x|}$ is close to such a point~$x_0\in S_{\nu-1}$
in whose small neighbourhood the function $a(\cdot)$ is 
separated from zero. Hence by an argument similar to the 
one in Part~(a) we get the inequality
$$
J_{\kappa,k,a}(x)\left\{
\begin{array}{l}
\ge \bar C(\kappa,k)|x|^{2\kappa k-\nu}
\quad\textrm{if }\frac\nu k>2\kappa>0,\\
=\infty\qquad \textrm{if }\kappa\le0 
\textrm{ or } 2\kappa\ge\frac\nu k
\end{array}\right.
$$
for such vectors $x\in R^\nu$.

Since $|\tilde\chi_n(x)|^2>0$ for almost all $x\in R^\nu$,
$$
D(n)=\int|\tilde \chi_n(x)|^2J_{\kappa,k,a}(x)\,dx=\infty
$$
under the conditions of Part~(b). Similarly $D(\varphi)=\infty$
if $|\tilde\varphi(x)|^2>0$ for almost all $x\in R^\nu$. We remark
that the conditions in Part~(b) can be weakened. It would have been
enough to assume that $a(x)>0$ on a set of positive Lebesgue
measure in~$S_{\nu-1}$.  \hfill$\qed$

\medskip
Theorem 6.2 and $6.2'$ together with Proposition 6.3 have the
following

\medskip\noindent
{\bf Corollary 6.4.} {\it The formulae
\begin{eqnarray*}
\xi_n=\sum_{k=1}^M \int\tilde\chi_n(x_1+\cdots+x_k)
&&\prod_{l=1}^k \left(|x_l|^{-\kappa+(\nu-\alpha)/k}
\cdot b_k\left(\frac{x_l}{|x_l|}\right)\right) \\
&&\qquad Z_G(\,dx_1)\dots Z_G(\,dx_k),
\qquad n\in\textrm{\BBB Z}_\nu,
\end{eqnarray*}
and
\begin{eqnarray*}
\xi(\varphi)=\sum_{k=1}^M \int\tilde\varphi(x_1+\cdots+x_k)
&&\prod_{l=1}^k\left(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot
b_k\left(\frac{x_l}{|x_l|}\right)\right) \\
&&\qquad Z_G(\,dx_1)\dots Z_G(\,dx_k), 
\qquad \varphi\in{\cal S},
\end{eqnarray*}
define self-similar random fields with self-similarity
parameter~$\alpha$ if $G$ is defined by formula~(\ref{(6.6)}), 
the parameter~$\alpha$ satisfies the inequality 
$\frac\nu2<\alpha<\nu$, and the functions $a(\cdot)$ (in 
the definition of the measure~$G(\cdot)$ in~(\ref{(6.6)})) and 
$b_1(\cdot)$,\dots, $b_k(\cdot)$ are bounded even 
functions on $S_{\nu-1}$.}

\medskip
The following observation may be useful in the proof of 
Corollary~6.4. We can replace $\xi_n$ by another random 
field with the same distribution. Thus we can write, by 
exploiting Theorem~4.5,
$$
\xi_n=\sum_{k=1}^M\tilde\chi_n(x_1+\cdots+x_k)
Z_{G'}(\,dx_1)\dots Z_{G'}(\,dx_k),
\quad n\in\textrm{\BBB Z}_\nu,
$$
with a random spectral measure $Z_{G'}$ corresponding to 
the spectral measure
$G'(\,dx)=b(\frac x{|x|})^2|x|^{-2\kappa+2(\nu-\alpha)/k}G(\,dx)
=a(\frac x{|x|})b(\frac x{|x|})^2|x|^{-\nu+2(\nu-\alpha)/k}\,dx$.
In the case of generalized random fields a similar 
argument can be applied.

\medskip\noindent
{\it Remark 6.5.} The estimate on $J_{\kappa,k}$ and the end of 
the of Part~(a) in Proposition~6.3 show that the self-similar 
random field
\begin{eqnarray*}
\xi(\varphi)&&=\sum_{k=1}^M\int \tilde\varphi(x_1+\cdots+x_k)
|x_1+\cdots+x_k|^p \, u
\left(\frac{x_1+\cdots+x_k}{|x_1+\cdots+x_k|}\right)\\
&&\qquad \prod_{l=1}^k\left(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot
b_k\left(\frac{x_l}{|x_l|}\right)\right) 
Z_G(\,dx_1)\dots Z_G(\,dx_k),
\quad \varphi\in{\cal S},
\end{eqnarray*}
and
\begin{eqnarray*}
\xi_n&&=\sum_{k=1}^M\int \tilde\chi_n(x_1+\cdots+x_k)
|x_1+\cdots+x_k|^p \, u
\left(\frac{x_1+\cdots+x_k}{|x_1+\cdots+x_k|}\right)\\
&&\qquad \prod_{l=1}^k\left(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot
b_k\left(\frac{x_l}{|x_l|}\right)\right) 
Z_G(\,dx_1)\dots Z_G(\,dx_k),
\quad n\in\textrm{\BBB Z}_\nu,
\end{eqnarray*}
are well defined if $G$ is defined by formula~(\ref{(6.6)}), 
$a(\cdot)$, $b(\cdot)$ and $u(\cdot)$ are bounded even 
functions on $S_{\nu-1}$, $\frac\nu2<\alpha<\nu$, and 
$\alpha-p<\nu$ in the generalized and
$\frac{\nu-1}2<\alpha-p<\nu$ is the discrete random field 
case. The self-similarity parameter of these random fields 
is $\alpha-p$. We remark that in the case $p>0$  this 
class of self-similar fields also contains self-similar 
fields with self-similarity parameter less than~$\frac\nu2$.

\medskip
In proving the statement of Remark~6.5 we have to check the
integrability conditions needed for the existence of the
Wiener--It\^o integrals $\xi(\varphi)$ and $\xi_n$. To check 
them it is worth remarking that in the proof of Part~(a) of
Proposition~6.3 we proved the estimate $J_{\bar\kappa,k}(x)\le
C(\bar\kappa,k)|x|^{2\bar\kappa k-\nu}$. We want to apply this
inequality in the present case with the choice
$\bar\kappa=\frac{\nu-\alpha}k$. Then arguing similarly to 
the proof of Part~(a) of Proposition~6.3 we get to the 
problem whether the relations 
$\int|\tilde\chi_n(x)|^2|x|^{2p+2(\nu-\alpha)-\nu}\,dx<\infty$
and 
$\int|\tilde\varphi(x)|^2|x|^{2p+2(\nu-\alpha)-\nu}\,dx<\infty$ 
if $\varphi\in{\cal S}$ hold under the conditions of 
Remark~6.5. They can be proved by means of the argument 
applied at the end of the proof of Part~(a) of Proposition~6.3.

\medskip
The following question arises in a natural way. When do different
formulas satisfying the conditions of Theorem~6.2 or Theorem~$6.2'$
define self-similar random fields with different distributions?
In particular: Are the self-similar random fields constructed 
via multiple Wiener--It\^o integrals of order $k\ge2$ necessarily 
non-Gaussian? We cannot give a completely satisfactory answer 
for the above question, but our former results yield some 
useful information. 

Let us substitute the spectral measure $G$ by $G'$ such that
$\frac{G(\,dx)}{G'(\,dx)}=|g^2(x)|^2$, $g(-x)=\overline{g(x)}$ and 
the functions $|x_l|^{-\kappa+(\nu-\alpha)/k}b(\frac{x_l}{|x_l|})$ 
by $b(\frac{x_l}{|x_l|})g(x_l)|x_l|^{-\kappa+(\nu-\alpha)/k}$ in
Corollary~6.4. By Theorem~4.4 the new field has the same distribution
as the original one. On the other hand, Corollary~5.4 may helps us 
to decide whether two random variables have different moments, and
therefore different distributions. Let us consider e.g. a moment 
of odd order of the random variables $\xi_n$ or $\xi(\varphi)$
defined in Corollary~6.4. It is clear that all $h_\gamma\ge0$.
Moreover, if $b_k(x)$ does not vanish for some even number~$k$, then
there exists a $h_\gamma>0$ in the sum expressing an odd moment of
$\xi_n$ or $\xi(\varphi)$. Hence the odd moments of $\xi_n$ or
$\xi(\varphi)$ are positive in this case. This means in particular
that the self-similar random fields defined in Corollary~6.4 are
non-Gaussian if $b_k$ is non-vanishing for some even~$k$. The next
result shows that the tail behaviour of multiple Wiener--It\^o
integrals of different order is different.

\medskip\noindent
{\bf Theorem 6.6.} {\it Let $G$ be a non-atomic spectral measure 
and $Z_G$ a random spectral measure corresponding to~$G$. For all 
$h\in{\cal H}_G^m$ there exist some constants $K_1>K_2>0$ and 
$x_0>0$ depending on the function~$h$ such that
$$
e^{-K_1x^{2/m}}\le P(|I_G(h)|>x)\le e^{-K_2x^{2/m}}
$$
for all $x>x_0$.}

\medskip\noindent
{\it Remark.}\/ As the proof of Theorem~6.6 shows the constant~$K_2$
in the upper bound of the above estimate can be chosen as
$K_m=C_m (EI_G(h)^2)^{-1/m}$ with a constant~$C_m$ depending only on
the order~$m$ of the Wiener--It\^o integral of~$I_G(h)$. This means
that for a fixed number~$m$ the constant~$K_2$ in the above estimate
can be chosen as a constant depending only on the variance of the
random variable~$I_G(h)$. On the other hand, no simple
characterization of the constant~$K_1>0$ appearing in the lower
bound of this estimate is known.

\medskip\noindent
{\it Proof of Theorem 6.6.} {\it (a) Proof of the upper estimate.}

We have
$$
P(|I_G(h)|>x)\le x^{2N}E(I_G(h)|^{2N}).
$$
By Corollary~5.6
$$
E(I_G(h)|^{2N})\le \bar C(m,N)[E(I_G(h)^2)]^N\le \bar C(m,N) C_1^N
$$
with the coefficient $\bar C(m,N)$ appearing in this result, 
and by a simple combinatorial argument we obtain that
$$
\bar C(m,N)\le\frac{(2Nm-1)(2Nm-3)\cdots 1}{(m!\,)^N},
$$
since the numerator on the right-hand side of this inequality
equals the number of complete diagrams
$|\bar\Gamma(\underbrace {m,\dots,m}_{2N \textrm{ times }})|$
 if vertices from the same row can also be connected. Multiplying the
inequalities
$$
(2nM-2j-1)(2Nm-2j-1-2N)\cdots (2Nm-2j-1-2N(m-1))\le (2N)^mm!,
$$
$j=1,\dots,N$, we obtain that
$$
\bar C(m,N)\le (2N)^{mN}.
$$
(This inequality could be sharpened, but it is sufficient for our
purpose.) Choose a sufficiently small number $\alpha>0$, and define
$N=[\alpha x^{2/m}]$, where $[\cdot]$ denotes integer part. With
this choice we have
$$
P(|I_G(h)|>x)\le(x^{-2}(2\alpha)^mx^2)^N C_1^N=[C_1(2\alpha)^m]^N
\le e^{-K_2x^{2/m}},
$$
if $\alpha$ is chosen in such a way that $C_1(2\alpha)^m\le\frac1e$,
$K_2=\frac\alpha2$, and $x>x_0$ with an appropriate $x_0>0$.

\medskip\noindent
{\it (b) Proof of the lower estimate.}

First we reduce this inequality to the following statement. Let
$Q(x_1,\dots,x_k)$ be a homogeneous polynomial of order~$m$ (the
number~$k$ is arbitrary), and $\xi=(\xi_1,\dots,\xi_k)$ a
$k$-dimensional standard normal variable. Then
\begin{equation}
P(Q(\xi_1,\dots,\xi_k)>x)\ge e^{-Kx^{2/m}} \label{(6.9)}
\end{equation}
if $x>x_0$, where the constants $K>0$ and $x_0>0$ may depend on the
polynomial~$Q$.

By the results of Chapter~4, $I_G(h)$ can be written in the form
\begin{equation}
I_G(h)=\sum_{j_1+\cdots+j_l=m} C^{k_1,\dots,k_l}_{j_1,\dots,j_l}
H_{j_1}(\xi_{k_1})\cdots H_{j_k}(\xi_{k_l}),
\label{(6.10)}
\end{equation}
where $\xi_1,\xi_2,\dots$ are independent standard normal 
random variables, $C^{k_1,\dots,k_l}_{j_1,\dots,j_l}$ are
appropriate coefficients, and the right-hand side of~(\ref{(6.10)}) 
is convergent in~$L_2$ sense. Let us fix a sufficiently large
integer~$k$, and let us consider the conditional distribution 
of the right-hand side of~(\ref{(6.10)}) under the condition
$\xi_{k+1}=x_{k+1},\xi_{k+2}=x_{k+2},\dots$, where the numbers
$x_{k+1},x_{k+2},\dots$ are arbitrary. This conditional 
distribution coincides with the distribution of the random variable
$Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)$ with probability~1,
where the polynomial $Q$ is obtained by substituting 
$\xi_{k+1}=x_{k+1}$, $\xi_{k+2}=x_{k=2},\dots$ into the 
right-hand side of~(\ref{(6.10)}). In particular, 
$$
Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)
$$ 
is a random polynomial with finite second moment, and as a 
consequence with finite coefficients for almost all 
vectors $(x_{k+1},x_{k+2},\dots)$ with respect
to the distribution of the vector $(\xi_{k+1},\xi_{k+2},\dots)$. 
It is clear that all these polynomials 
$$
Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)
$$ 
are of order $m$ if $k$ is sufficiently large. It is sufficient 
to prove that
$$
P(|Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)|>x)\ge e^{-Kx^{2/m}}
$$
for $x>x_0$, where the constants $K>0$ and $x_0>0$ may depend on
the polynomial~$Q$. Write
$$
Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)=
Q_1(\xi_1,\dots,\xi_k)+Q_2(\xi_1,\dots,\xi_k)
$$
where $Q_1$ is a homogeneous polynomial of order~$m$, and~$Q_2$ is a
polynomial of order less than~$m$. The polynomial $Q_2$ can be
rewritten as the sum of finitely many Wiener--It\^o integrals with
multiplicity less than~$m$. Hence the already proved part of
Theorem~6.6 implies that
$$
P(Q_2(\xi_1,\dots,\xi_k)>x)\le e^{-\bar qKx^{2/(m-1)}} .
$$
(We may assume that $m\ge2$). Then an application of 
relation~(\ref{(6.9)})
to~$Q_1$ implies the remaining part of Theorem~6.6, thus it
suffices to prove~(\ref{(6.9)}).

\medskip
If $Q(\cdot)$ is a polynomial of $k$ variables, then there exist
some $\alpha>0$ and $\beta>0$ such that
$$
\lambda\left(\left|Q\left(\frac{x_1}{|x|},\dots,
\frac{x_k}{|x|}\right)\right|>\alpha\right)>\beta,
$$
where $|x|^2=\sum\limits_{j=1}^kx_j^2$, and $\lambda$ denotes the 
Lebesgue measure on the $k$-dimensional unit sphere $S_{k-1}$. 
Exploiting that $|\xi|$ and $\frac\xi{|\xi|}$ are independent, 
$\frac\xi{|\xi|}$ is uniformly distributed on the unit sphere 
$S_{k-1}$, and $P(|\xi|>x)\ge ce^{-x^2}$ for a $k$-dimensional 
standard normal random variable, we obtain that
$$
P(|Q(\xi_1,\dots,\xi_k)|>x)\ge\beta P\left(|\xi|^m>
\frac x\alpha\right)>e^{-Kx^{2/m}},
$$
if the constants $K$ and $x$ are sufficiently large. Theorem~6.6 is
proved. \hfill$\qed$

\medskip
Theorem~6.6 implies in particular that Wiener--It\^o integrals of
different multiplicity have different distributions. A bounded
random variable measurable with respect to the $\sigma$-algebra 
generated by a stationary Gaussian field can be expressed as a sum of
multiple Wiener--It\^o integrals. Another consequence of Theorem~6.6
is the fact that the number of terms in this sum must be infinite.

In Theorems~6.2 and~$6.2'$ we have defined a large class of
self-similar fields. The question arises whether this class contains
self-similar fields such that the distributions of their random
variables tend to  one (or zero) at infinity (at minus infinity)
much faster than the normal distribution functions do. This
question has been unsolved by now. By Theorem~6.6 such fields, if
any, must be expressed as a sum of infinitely many Wiener--It\^o
integrals. The above question is of much greater importance
than it may seem at first instant. Some considerations suggest
that in some important models of statistical physics self-similar
fields with very fast decreasing tail distributions appear as
limit, when the so-called renormalization group transformations are
applied for the probability measure describing the state of the
model at critical temperature. (The renormalization group
transformations are the transformations over the distribution of
stationary fields induced by formula~(\ref{(1.1)}) or~(\ref{(1.3)}),
when $A_N=N^\alpha$, $A(t)=t^\alpha$ with some~$\alpha$.) No
rigorous proof about the existence of such self-similar fields is
known yet. Thus the real problem behind the above question is whether
the self-similar fields interesting for statistical physics can be
constructed via multiple Wiener--It\^o integrals.

\chapter{On the Original Wiener--It\^o Integral}

In this chapter the definition of the original Wiener--It\^o
integral introduced by It\^o in~\cite{r18} is explained. As the
arguments are very similar to those of Chapters~4 and~5 (only the
notations become simpler) most proofs will be omitted.

Let a measure space $(M,{\cal M},\mu)$ with a $\sigma$-finite
measure~$\mu$ be given. Let $\mu$ satisfy the following continuity
property: For all $\varepsilon>0$ and $A\in{\cal M}$, 
$\mu(A)<\infty$, there exist some disjoint sets $B_j\in{\cal M}$, 
$j=1,\dots,N$, with some integer~$N$ such that 
$\mu(B_j)<\varepsilon$ for all $1\le j\le N$, and
$A=\bigcup\limits_{j=1}^NB_j$. We introduce the following definition.

\medskip\noindent
{\bf Definition of (Gaussian) Random Orthogonal Measures.} 
\index{Gaussian random orthogonal measure} 
{\it A system of random variables $Z_\mu(A)$, $A\in{\cal M}$,
$\mu(A)<\infty$, is called a Gaussian random orthogonal measure
corresponding to  the measure~$\mu$ if

\medskip
\begin{description}
\item[(i)] $Z_\mu(A_1),\dots,Z_\mu(A_k)$ are independent Gaussian
random variables if the sets $A_j\in{\cal M}$, $\mu(A_j)<\infty$,
$j=1,\dots,k$, are disjoint.
\item[(ii)] $EZ_\mu(A)=0$, $EZ_\mu(A)^2=\mu(A)$.
\item[(iii)] $Z_\mu\left(\bigcup\limits_{j=1}^k A_j\right)
=\sum\limits_{j=1}^k Z_\mu(A_k)$
with probability~1 if $A_1,\dots,A_k$ are disjoint sets.
\end{description}

}

\medskip\noindent
{\it Remark.}\/ There is the following equivalent version for the
definition of random orthogonal measures: The system of random variables
system of random variables $Z_\mu(A)$, $A\in{\cal M}$, $\mu(A)<\infty$, is
a Gaussian random orthogonal measure corresponding to  the
measure~$\mu$ if

\medskip
\begin{description}
\item[(i$'$)] $Z_\mu(A_1),\dots,Z_\mu(A_k)$ are (jointly) Gaussian
random variables for all sets $A_j\in{\cal M}$, $\mu(A_j)<\infty$,
$j=1,\dots,k$.
\item[(ii$'$)] $EZ_\mu(A)=0$, and $EZ_\mu(A)Z_\mu(B)=\mu(A\cap B)$
if $A,\,B\in{\cal M}$, $\mu(A)<\infty$, $\mu(B)<\infty$.
\end{description}

\medskip
It is not difficult to see that properties~(i),~(ii) and~(iii) 
imply relations~(i$'$) and~(ii$'$). On the other hand, it is 
clear that (i$'$) and (ii$'$) imply~(i) and~(ii). To see that 
they also imply relation~(iii) observe that under these 
conditions
$$
E\left[Z_\mu\left(\bigcup\limits_{j=1}^k A_j\right)
-\sum\limits_{j=1}^k Z_\mu(A_k)\right]^2=0
$$
if $A_1,\dots,A_k$ are disjoint sets.

The second characterization of random orthogonal measures may 
help to show that for any measure space $(M,{\cal M},\mu)$ 
with a $\sigma$-finite measure~$\mu$ there exists a Gaussian 
random orthogonal measure corresponding to  the measure~$\mu$. 
The main point in checking this statement is the proof that 
for any sets $A_1,\dots,A_k\in{\cal M}$, $\mu(A_j)<\infty$, 
$1\le j\le k$, there exists a Gaussian random vector 
$(Z_\mu(A_1),\dots,Z_\mu(A_k))$, $EZ_\mu(A_j)=0$, with 
correlation  $EZ_\mu(A_i)Z_\mu(A_j)=\mu(A_i\cap A_j)$ for 
all $1\le i,j\le k$.  To prove this we have to show that 
the corresponding covariance matrix is really positive definite, 
i.e. $\sum\limits_{i,j} c_i\bar c_j\mu(A_i\cap A_j)\ge0$ for 
an arbitrary vector $(c_1,\dots,c_k)$. But this follows from 
the observation $\sum\limits_{i,j} c_i\bar c_j\chi_{A_i\cap A_j}(x)
=\sum\limits_{i,j} c_i\bar c_j\chi_{A_i}(x)\overline{\chi_{ A_j}(x)}
=\left|\sum\limits_i c_i\chi_{A_i}(x)\right|^2\ge0$ for all 
$x\in M$, if we integrate this inequality with respect to 
the measure~$\mu$ in the space~$M$.

\medskip
We define the real Hilbert spaces $\bar{{\cal  K}}^n_\mu$, $n=1,2,\dots$.
The space $\bar{{\cal K}}^n_\mu$ consists of the real-valued measurable
functions over
$(\underbrace{M\times\cdots\times M}_{n\textrm{ times}},\,
\underbrace{{\cal M}\times\cdots\times{\cal M}}_{n\textrm{ times}})$
such that
$$
\|f\|^2=\int|f(x_1,\dots,x_n)|^2\mu(\,dx_1)\dots\mu(\,dx_n)<\infty,
$$
and the last formula defines the norm in $\bar{{\cal K}}^n_\mu$. 
Let ${\cal K}^n_\mu$ denote the subspace of $\bar{{\cal K}}^n_\mu$ 
consisting of the functions $f\in\bar{{\cal K}}^n_\mu$ such that
$$
f(x_1,\dots,x_n)=f(x_{\pi(1)},\dots,x_{\pi(n)})
\quad\textrm{for all }\pi\in\Pi_n.
$$
Let the spaces $\bar{{\cal K}}^0_\mu$ and ${\cal K}^0_\mu$ consist 
of the real constants with the norm $\|c\|=|c|$. Finally we 
define the Fock space $\textrm{Exp}\,{\cal K}_\mu$ which consists 
of the sequences $f=(f_0,f_1,\dots)$, $f_n\in{\cal K}^n_\mu$, 
$n=0,1,2,\dots$, such that
$$
\|f\|^2=\sum_{n=0}^\infty \frac1{n!} \|f_n\|^2<\infty.
$$

Given a random orthogonal measure $Z_\mu$ corresponding 
to $\mu$, let us introduce the $\sigma$-algebra
${\cal F}=\sigma(Z_\mu(A)\colon\; A\in{\cal M},\,\mu(A)<\infty)$. 
Let ${\cal K}$ denote the real Hilbert space of square 
integrable random variables measurable with respect to 
the $\sigma$-algebra~${\cal F}$. Let ${\cal K}_{\le n}$ 
denote the subspace of ${\cal K}$ that is the closure of the 
linear space containing the polynomials of the random variables 
$Z_\mu(A)$ of order less than or equal to~$n$. Let ${\cal K}_n$ 
be the orthogonal completion of ${\cal K}_{\le n-1}$ to 
${\cal K}_{\le n}$. (The norm is defined as $\|\xi\|^2=E\xi^2$ 
in these Hilbert spaces.)

The multiple Wiener--It\^o integrals with respect to the random
orthogonal measure $Z_\mu$, 
\index{Wiener--It\^o integrals with respect to a random orthogonal 
measure}
to be defined below, give a unitary transformation from 
$\textrm{Exp}\,{\cal K}_\mu$ to ${\cal K}$. We shall denote 
these integrals by $\int'$ to distinguish them from the
Wiener--It\^o integrals defined in Chapter~4.

First we define the class of simple functions
$\hat{\bar{{\cal K}}}_\mu^n\subset\bar{{\cal K}}_\mu^n$. 
A function $f\in\bar{{\cal K}}_\mu^n$ is in 
$\hat{\bar{{\cal K}}}_\mu^n$ if there exists a finite 
system of disjoint sets $\Delta_1,\dots,\Delta_N$,
with $\Delta_j\in{\cal M}$, $\mu(\Delta_j)<\infty$, \ 
$j=1,\dots,N$, such that $f(x_1,\dots,x_n)$ is constant 
on the sets $\Delta_{j_1}\times\cdots\times\Delta_{j_n}$ 
if the indices $j_1,\dots,j_n$ are disjoint, and 
$f(x_1,\dots,x_n)$ equals zero outside these sets. We define
\index{simple function}
$$
\int' f(x_1,\dots,x_n)Z_\mu(\,dx_1)\dots Z_\mu(\,dx_n)
=\sum f(x_{j_1},\dots,x_{j_n})Z_\mu(\Delta_{j_1})
\cdots Z_\mu(\Delta_{j_n})
$$
for $f\in\hat{\bar{{\cal K}}}_\mu^n$, where $x_k\in\Delta_k$, 
$k=1,\dots,N$. Here again, it can be seen with the help of the
additivity property~(iii) of the random orthogonal measure
$Z_\mu$ that the above definition of the Wiener--It\^o integral
of simple functions is meaningful, although the simple function
$f$ does not determine uniquely the sets $\Delta_j$ appearing
in this definition. 

Let $\hat{{\cal K}}_\mu^n=\hat{\bar{{\cal K}}}_\mu^n\cap{\cal K}_\mu^n$. 
The random variables
$$
I'_\mu(f)=\frac1{n!}\int'f(x_1,\dots,x_n)
Z_\mu(\,dx_1)\dots Z_\mu(\,dx_n),
\quad f\in\hat{\bar{{\cal K}}}_\mu^n,
$$
have zero expectation, integrals of different order are orthogonal,
$$
I'_\mu(f)=I'_\mu(\,\textrm{Sym}\, f), \quad\textrm{and }
\textrm{Sym}\,f\in \hat{{\cal K}}_\mu^n \textrm{ if }
f\in\hat{\bar{{\cal K}}}_\mu^n,
$$
\begin{equation}
EI'_\mu(f)^2\le \frac1{n!}\|f\|^2 \quad \textrm{if }
f\in\hat{\bar{{\cal K}}}_\mu^n, \label{(7.1)}
\end{equation}
and~(\ref{(7.1)}) holds with equality if 
$f\in\hat{{\cal K}}_\mu^n$.

It can be seen that $\hat{\bar{{\cal K}}}^n_\mu$ is dense
in~$\bar{{\cal K}}_\mu^n$ in the $L_2(\mu^n)$ norm. (This
is a statement analogous to Lemma~4.1, but its proof is
simpler.) Hence relation~(\ref{(7.1)})  enables us to 
extend the definition of the $n$-fold Wiener--It\^o 
integrals over~$\bar{{\cal K}}_\mu^n$. All the above 
mentioned relations  remain valid if 
$f\in\hat{\bar{{\cal K}}}_\mu^n$ is substituted by 
$f\in\bar{{\cal K}}_\mu^n$, and 
$f\in\hat{{\cal K}}_\mu^n$ is substituted by 
$f\in{\cal K}_\mu^n$. We formulate It\^o's  formula 
for these integrals. It can be proved similarly to 
Theorem~4.3 with the help of the diagram formula valid
for the classical Wiener--It\^o integrals studied in this
chapter. 

\medskip\noindent
{\bf Theorem~7.1. (It\^o's Formula.)} 
\index{It\^o's formula for Wiener--It\^o integrals with respect 
to a random orthogonal measure}
{\it Let
$\varphi_1,\dots,\varphi_m$, $\varphi_j\in{\cal K}^1_\mu$ for all
$1\le j\le m$, be an orthonormal system in $L^2_\mu$. Let some
positive integers $j_1,\dots,j_m$ be given, put $j_1+\cdots+j_m=N$,
and define for all $i=1,\dots,N$
$$
g_i=\varphi_1 \textrm{ for } 1\le i\le j_1,\quad\textrm{and \ }
g_i=\varphi_s \quad\textrm{for } j_1+\cdots+j_{s-1}<i\le j_1+\cdots+j_s.
$$
Then
\begin{eqnarray*}
&&H_{j_1}\left(\int'\varphi_1(x)Z_\mu(\,dx)\right)\cdots
H_{j_m}\left(\int'\varphi_m(x)Z_\mu(\,dx)\right)\\
&&\qquad=\int' g_1(x_1)\cdots g_N(x_N)\,Z_\mu(\,dx_1)\dots Z_\mu(\,dx_N)\\
&&\qquad=\int'\textrm{\rm Sym}\,
[ g_1(x_1)\cdots g_N(x_N)]\,Z_\mu(\,dx_1)\dots Z_\mu(\,dx_N).
\end{eqnarray*}
}

\medskip
(Let me remark that the diagram formula (Theorem~5.3) also remains
valid for this integral if we replace $-x_j$ is by $x_j$ and
$G(\,dx_j)$ by $\mu(\,dx_j)$, $N-2|\gamma|+1\le j\le N-|\gamma|$,
in the definition of~$h_\gamma$ in formula~(\ref{(5.1)}).)

\medskip
It can be seen with the help of ~Theorem~7.1 that the transformation
$$
I'_\mu\colon\;\textrm{Exp}\,{\cal K}_\mu\to{\cal K},
$$ 
where $I'_\mu(f)=\sum\limits_{n=0}^\infty I'_\mu(f_n)$, \
$f=(f_0,f_1,\dots)\in\,\textrm{Exp}\,{{\cal K}}_\mu$ is a unitary
transformation, and so are the transformations $(n!)^{1/2}I'_\mu$
from ${\cal K}_\mu^n$ to~${\cal K}_n$.

\medskip
Let us consider the special case
$(M,{\cal M},\mu)=(R^\nu,{\cal B}^\nu,\lambda)$, where $\lambda$ 
denotes the Lebesgue measure in~$R^\nu$. A random orthogonal measure
corresponding to $\lambda$ is called the white noise. A random
{\it spectral measure}\/ corresponding to $\lambda$, when the
Lebesgue measure is considered as the spectral measure of a
generalized field, is also called a white noise. The next result,
that can be considered as a random Plancherel formula, establishes
a connection between the two types of Wiener--It\^o integrals with
respect to white noise.
\index{white noise}

\medskip\noindent
{\bf Proposition 7.2.} {\it Let
$f=(f_0,f_1,\dots,)\in\,\textrm{\rm Exp}\,{\cal K}_\lambda$ be 
an element of the Fock space corresponding to the Lebesgue 
measure in the Euclidean space $(R^\nu,{\cal B}^\nu)$. Then
$f'=(f'_0,f'_1,\dots,)\in\,\textrm{\rm Exp}\,{\cal H}_\lambda$ 
with the functions 
$f'_0=f_0$ and $f'_n=(2\pi)^{-n\nu/2}\tilde f_n$,
$n=1,2,\dots$, (where
$\tilde f_n(u_1,\dots,u_n)=
\int_{R^{n\nu}}e^{i(x,u)}f_n(x_1,\dots,x_n)\,dx_1\dots\,dx_n$ with
$x=(x_1,\dots,x_n)$ and $u=(u_1,\dots,u_n)$),
and
\begin{eqnarray*}
&&\sum_{n=0}^\infty \frac1{n!}\int' f_n(x_1,\dots,x_n)\,
Z_\lambda(\,dx_1)\dots Z_\lambda(\,dx_n) \\
&&\qquad \stackrel{\Delta}{=}\sum_{n=0}^\infty \frac1{n!}
\int f'_n(u_1,\dots,u_n)\, Z_\lambda(\,du_1)\dots Z_\lambda(\,du_n),
\end{eqnarray*}
\index{Fock space}
where $Z_\lambda(\,dx)$ is a white noise as a random orthogonal
measure, and $Z_\lambda(\,du)$ is a white noise as a random spectral
measure.}

\medskip\noindent
{\it Proof of Proposition 7.2.}\/ We have
$$
(2\pi)^{-n\nu/2}\|\tilde f_n\|_{L_\lambda^2}=\|f_n\|_{L_\lambda^2},
$$
hence $f'\in\,\textrm{Exp}\,{\cal H}_\lambda$.

Let $\varphi_1,\varphi_2,\dots$ be a complete orthonormal system
in~$L_\lambda^2$. Then $\varphi'_1,\varphi'_2,\dots$ is also a
complete orthonormal system in $L_\lambda^2$, and if
$$
f_n(x_1,\dots,x_n)
=\sum c_{j_1,\dots,j_n}\varphi_{j_1}(x_1)\cdots\varphi_{j_n}(x_n),
$$
then
$$
f'_n(u_1,\dots,u_n)
=\sum c_{j_1,\dots,j_n}\varphi'_{j_1}(u_1)\cdots\varphi'_{j_n}(u_n).
$$
Hence an application of It\^o's formula for both types of integrals,
(i.e. Theorems~4.3 and~7.1) imply Proposition~7.2.
\hfill$\qed$

\medskip\noindent
Finally we restrict ourselves to the case $\nu=1$. We formulate a
result which reflects a connection between multiple Wiener--It\^o
integrals and classical It\^o integrals. Let $W(t)$,
$a\le t\le b$, be a Wiener process, and let us define the random
orthogonal measure $Z(\,dx)$ as
$$
Z(A)=\int\chi_A(x)W(\,dx), \quad A\subset[a,b),\quad A\in{\cal B}^1.
$$
Then we have the following

\medskip\noindent
{\bf Proposition 7.3.} {\it Let $f\in{\cal K}^n_{\lambda[a,b)}$, 
where $\lambda[a,b)$ denotes the Lebesgue measure on the 
interval~$[a,b)$. Then
\begin{eqnarray}
&&\int' f(x_1,\dots,x_n)\,Z(\,dx_1)\dots Z(\,dx_n) \label{(7.2)} \\
&&\qquad=n!\int_a^b \left(\int_a^{t_n}\left(\cdots 
\left(\int_a^{t_3}\left(\int_a^{t_2}
f(t_1,\dots,t_n)W(\,dt_1)\right)W(\,dt_2)\right)\dots\right) 
W(\,dt_n)\right). \nonumber
\end{eqnarray}
}

\medskip\noindent
{\it Proof of Proposition 7.3.}\/ Given a function
$f\in\hat{{\cal K}}^n_{\lambda[a,b)}$, let the function $\hat f$ be
defined as
$$
\hat f(x_1,\dots,x_n)=\left\{
\begin{array}{l}
f(x_1,\dots,x_n)\quad\textrm{if \ }x_1<x_2<\cdots<x_n \\
0 \quad \textrm{otherwise.}
\end{array} \right.
$$
It is not difficult to check Proposition~7.3 for such a 
special function $f\in\hat{{\cal K}}^n_{\lambda[a,b)}$ 
for which the above defined function $\hat f$ is the 
indicator function of a rectangle of the form 
$\prod\limits_{j=1}^n[a_j,b_j)$ with constants
$a\le a_1<b_1<a_2<b_2<\cdots<a_n<b_n\le b$. Here we 
exploit the relation $I'(f)=n!I'(\hat f)$. Beside 
this, we have to calculate the value of the right-hand 
side of formula~(\ref{(7.2)}) for these simple 
functions~$f\in\hat{{\cal K}}^n_{\lambda[a,b)}$. 
A simple inductive argument shows that it equals
$\prod\limits_{j=1}^n[W(b_j)-W(a_j)]$ if
$a\le a_1<b_1<a_2<b_2<\cdots<a_n<b_n\le b$, and it 
equals zero otherwise. Then a simple limiting procedure 
with the help of the approximation of general functions 
in ${\cal K}^n_{\lambda[a,b)}$ by the linear combinations 
of such functions proves Proposition~7.3 in the general 
case. \hfill$\qed$

\medskip
As a consequence of Proposition~7.3 in the case $\nu=1$ 
multiple Wiener--It\^o integrals can be substituted by 
It\^o integrals in the investigation of most problems. 
In the case $\nu=2$ there is no simple definition of 
It\^o integrals. On the other hand, no problem arises 
in generalizing the definition of multiple 
Wiener--It\^o integrals to the case $\nu\ge2$.

\chapter{Non-central Limit Theorems}

In this chapter we investigate the problem formulated in 
Chapter~1, and we show how the technique of Wiener--It\^o 
integrals can be applied for the investigation of such a 
problem. We restrict ourselves to the case of discrete 
random fields, although the case of generalized random 
fields can be discussed in almost the same way. We also 
present some generalizations of these results which can 
be proved in a similar way. But the proof of these more
general results will be omitted. They can be found 
in~\cite{r9}. First we recall the following

\medskip\noindent
{\bf Definition 8A. (Definition of Slowly Varying Functions.)} 
\index{slowly varying function} 
{\rm A function $L(t)$, $t\in[t_0,\infty)$, $t_0>0$, is 
said to be a slowly varying function (at infinity) if
$$
\lim_{t\to\infty}\frac{L(st)}{L(t)}=1 
\quad\textrm{for all \ } s>0.
$$
}

\medskip
We shall apply the following description of slowly 
varying functions.

\medskip\noindent
{\bf Theorem 8A. (Karamata's Theorem.)} 
\index{Karamata's theorem} 
{\it If a slowly varying function $L(t)$, $t\ge t_0$,
with some $t_0>0$, is bounded on every 
finite interval, then it can be represented in the form
$$
L(t)=a(t)\exp\left\{\int_{t_0}^t 
\frac{\varepsilon(s)}s\,ds\right\},
$$
where $a(t)\to a_0\neq0$, $\varepsilon(t)$ is integrable
on any finite intervals $[t_0,t]$, and $\varepsilon(t)\to0$ as 
$t\to\infty$.} 

\medskip
Let $X_n$, $n\in\,\textrm{\BBB Z}_\nu$, be a stationary 
Gaussian field with expectation zero and  a 
correlation function
\begin{equation}
r(n)=EX_0X_n=|n|^{-\alpha}a\left(\frac n{|n|}\right)L(|n|),
\quad n\in\textrm{\BBB Z}_\nu, \quad \textrm{if }n\neq(0,\dots,)), 
\label{(8.1)}
\end{equation}
where $0<\alpha<\nu$, $L(t)$, $t\ge1$, is a slowly varying 
function, bounded in all finite intervals, and 
$a(t)$ is a continuous function on the unit sphere 
${\cal S}_{\nu-1}$, satisfying the symmetry property 
$a(x)=a(-x)$ for all $x\in{\cal S}_{\nu-1}$. Let $G$ 
denote the spectral measure of the field~$X_n$, and 
let us define the measures~$G_N$, $N=1,2,\dots$, 
by the formula
\begin{equation}
G_N(A)=\frac{N^\alpha}{L(N)}G\left(\frac AN\right),
\quad A\in{\cal B}^\nu, \quad N=1,2,\dots. \label{(8.2)}
\end{equation}

Now we recall the definition of vague convergence of 
not necessarily finite measures on a Euclidean space.

\medskip\noindent
{\bf Definition of Vague Convergence of Measures.} 
\index{vague convergence of measures} 
{\it Let $G_n$, $n=1,2,\dots$, be a sequence of locally 
finite measures over $R^\nu$, i.e. let $G_n(A)<\infty$ 
for all measurable bounded sets~$A$. We say that the 
sequence $G_n$ vaguely converges to a locally finite
measure~$G_0$ on~$R^\nu$ (in notation 
$G_n\stackrel{v}{\rightarrow} G_0$) if
$$
\lim_{n\to\infty}\int f(x)\,G_n(\,dx)=\int f(x)\,G_0(\,dx)
$$
for all continuous functions~$f$ with a bounded support.}

\medskip
We formulate the following

\medskip\noindent
{\bf Lemma 8.1.} {\it Let $G$ be the spectral measure of 
a stationary random field with a correlation function 
$r(n)$ of the form~(\ref{(8.1)}). Then the sequence of 
measures~$G_N$ defined in~(\ref{(8.2)}) tends vaguely to 
a locally finite measure~$G_0$. The measure~$G_0$ has 
the homogeneity property
\begin{equation}
G_0(A)=t^{-\alpha}G_0(tA) \quad \textrm{for all } A\in{\cal B}^\nu
\quad\textrm{and } t>0, \label{(8.3)}
\end{equation}
and it satisfies the identity
\begin{eqnarray}
&&2^\nu\int e^{i(t,x)}
\prod_{j=1}^\nu\frac{1-\cos x^{(j)}}{(x^{(j)})^2}
\,G_0(\,dx) \label{(8.4)} \\
&&\qquad =\int_{[-1,1]^\nu} (1-|x^{(1)}|)\cdots (1-|x^{(\nu)}|)
\frac{a\left(\frac{x+t}{|x+t|}\right)}{|x+t|^\alpha}\,dx, 
\quad\textrm{for all } t\in R^\nu. \nonumber
\end{eqnarray}
}

\medskip\noindent
{\it Remark.} One may ask whether there are stationary
random fields with correlation function satisfying 
relation~(\ref{(8.1)}), or more generally, how large
the class of such random fields is. It can be proved that
we get a correlation function of the form~(\ref{(8.1)})
with the help of a spectral measure with a spectral 
density of the form 
$g(u)=|u|^{\alpha-\nu}b(\frac u{|u|})h(|u|)$, $u\in R^\nu$, 
where $b(\cdot)$ is a non-negative smooth function on the 
unit sphere $\{u\colon\; u\in R^\nu,\,|u|=1\}$, and $h(u)$ 
is a non-negative smooth function on $R^1$ which does not 
disappear at the origin, and tends to zero at infinity 
sufficiently fast. The regularizing function $h(|u|)$ is 
needed in this formula to make the function $g(\cdot)$ 
integrable. Results of this type are studied in the theory 
of generalized functions.

At a heuristic level the class of spectral measures
$G(\cdot)$ which determine a correlation function $r(\cdot)$
satisfying relation~(\ref{(8.1)}) can be described in the
following way. They are such measures~$G$ for which the 
asymptotic identity $G(B_x)\sim C x^\alpha$ holds with 
some constant $C>0$ for the (small) balls 
$B_x=\{v\colon\; |v|\le x\}$ as $x\to0$, and the 
effect of the singularities of the measure $G$ outside 
the origin is in some sense small. At this heuristic 
level we disregarded the possibility of a factor $L(|x|)$  
with a function $L(\cdot)$, slowly varying at the origin. 
Thus heuristically we can say that the asymptotically 
homogeneous behaviour $r(n)\sim Cn^{-\alpha}$ 
of the correlation function at infinity corresponds to 
the asymptotically homogeneous behaviour 
$G(B_x)\sim\bar Cx^{\alpha}$ of the spectral measure~$G$ 
corresponding to it in the neighbourhood of zero together 
with some additional restrictions about the singularities
of the spectral measure $G$ outside zero which 
guarantee that their influence is not too strong. These 
considerations may help us to understand the content of 
one of the most important conditions in the subsequent 
Theorem~8.2.

\medskip
We postpone the proof of Lemma~8.1 for a while.

Formulae~(\ref{(8.3)}) and~(\ref{(8.4)}) imply that the 
function~$a(t)$ and the number~$\alpha$ in the 
definition~(\ref{(8.1)}) of a correlation function~$r(n)$ 
uniquely determine the measure~$G_0$. Indeed, by 
formula~(\ref{(8.4)}) they determine the (finite) measure
$\prod\limits_{j=1}^\nu\frac{1-\cos x^{(j)}}{(x^{(j)})^2}G_0(\,dx)$, 
since they determine its Fourier transform. Hence they also 
determine the measure~$G_0$. (Formula~(\ref{(8.3)}) shows 
that $G_0$ is a locally finite measure). Let us also remark 
that since $G_N(A)=G_N(-A)$ for all $N=1,2,\dots$ and 
$A\in {\cal B}^\nu$, the relation $G_0(A)=G_0(-A)$,
$A\in{\cal B}^\nu$ also holds. These properties of the 
measure~$G_0$ imply that it can be considered as the 
spectral measure of a generalized random field. Now we 
formulate

\medskip\noindent
{\bf Theorem 8.2.} {\it Let $X_n$, $n\in\textrm{\BBB Z}_\nu$, be 
a stationary Gaussian field with a correlation function $r(n)$
satisfying relation~(\ref{(8.1)}). Let us define the stationary 
random field $\xi_j=H_k(X_j)$, $j\in\textrm{\BBB Z}_\nu$, with 
some positive integer~$k$, where $H_k(x)$ denotes the $k$-th 
Hermite polynomial with leading coefficient~1, and assume that 
the parameter~$\alpha$ appearing in~(\ref{(8.1)}) satisfies 
the relation $0<\alpha<\frac\nu k$ with this number~$k$. If 
the random fields $Z^N_n$, $N=1,2,\dots$, 
$n\in\textrm{\BBB Z}_\nu$, are defined by formula~(\ref{(1.1)}) 
with $A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$ and the above defined 
$\xi_j=H_k(X_j)$, then their multi-dimensional
distributions tend to those of the random field~$Z^*_n$,
$$
Z^*_n=\int \tilde\chi_n(x_1+\cdots+x_k)\,
Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k), 
\quad n\in\textrm{\BBB Z}_\nu.
$$
Here $Z_{G_0}$ is a random spectral measure corresponding to the
spectral measure $G_0$ which appeared in Lemma~8.1. The function
$\tilde\chi_n(\cdot)$, $n=(n^{(1)},\dots,n^{(\nu)})$, is (similarly
to formula~(\ref{(6.1a)}) Chapter~6) the Fourier transform of 
the indicator function of the $\nu$-dimensional unit cube
$\prod\limits_{p=1}^\nu[n^{(p)},n^{(p)}+1]$.}

\medskip\noindent
{\it Remark.}\/ The condition that the correlation function~$r(n)$
of the random field $X_n$, $n\in\textrm{\BBB Z}_\nu$, satisfies
formula~(\ref{(8.1)}) can be weakened. Theorem~8.2 and Lemma~8.1 remain
valid if~(\ref{(8.1)}) is replaced by the slightly weaker condition
$$
\lim_{T\to\infty}\sup_{n\colon\;n\in\textrm{\BBB Z}_\nu,\,|n|\ge T}
\frac{r(n)}{|n|^{-\alpha}a\left(\frac n{|n|}\right)L(|n|)}=1,
$$
where $0<\alpha<\nu$, $L(t)$, $t\ge1$, is a slowly varying function, 
bounded in all finite intervals, and $a(t)$ is a continuous function 
on the unit sphere ${\cal S}_{\nu-1}$, satisfying the symmetry property
$a(x)=a(-x)$ for all $x\in{\cal S}_{\nu-1}$.

\medskip
First we explain why the choice of the normalizing constant~$A_N$ in
Theorem~8.2 was natural, then we explain the ideas of the proof,
finally we work out the details.

It can be shown, for instance with the help of Corollary~5.5, that
$EH_k(\xi)H_k(\eta)=E\colon\!\xi^k\!\colon\colon\!\eta^k\!\colon
= k!(E\xi\eta)^k$ for a Gaussian random vector
$(\xi,\eta)$ with $E\xi=E\eta=0$ and $E\xi^2=E\eta^2=1$. Hence
$$
E(Z_n^N)^2=\frac{k!}{A_N^2}\sum_{j,\,l\in B_0^N}r(j-l)^k
\sim\frac{k!}{A_N^2}
\sum_{j,\,l\in B_0^N}|j-l|^{-k\alpha}a^k
\left(\frac{j-l}{|j-l|}\right)L(|j-l|)^k,
$$
with the set $B_0^N$ introduced after formula~(\ref{(1.1)}). Some 
calculation with the help of the above formula shows that with 
our choice of~$A_N$ the expectation $E(Z_n^N)^2$ is separated 
both from zero and infinity, therefore this is the natural 
norming factor. In this calculation we have to exploit the 
condition $k\alpha<\nu$, which implies that in the sum 
expressing $E(Z_n^N)^2$ those terms are dominant for which 
$j-l$ is relatively large, more explicitly which are of 
order~$N$. There are $\textrm{const.}\, N^{2\nu}$ such terms.

The field $\xi_n$, $n\in\textrm{\BBB Z}_\nu$, is subordinated 
to the Gaussian field~$X_n$. It is natural to write up its 
canonical representation defined in Chap.~6, and to express 
$Z_n^N$ via multiple  Wiener--It\^o integrals. It\^o's formula 
yields the relation
$$
\xi_j=H_k\left(\int e^{i(j,x)}Z_G(\,dx)\right)
=\int e^{i(j,x_1+\cdots+x_k)}Z_G(\,dx_1)\dots Z_G(\,dx_k),
$$
where $Z_G$ is the random spectral measure adapted to the random
field~$X_n$. Then
\begin{eqnarray*}
Z_n^N&=&\frac1{A_N}\sum_{j\in B_n^N}\int e^{i(j,x_1+\cdots+x_k)}
Z_G(\,dx_1)\dots Z_G(\,dx_k)\\
&=&\frac1{A_N}\int e^{i(Nn,x_1+\cdots+x_k)}\prod_{j=1}^\nu
\frac{e^{iN(x_1^{(j)}+\cdots+x_k^{(j)})}-1}
{e^{i(x_1^{(j)}+\cdots+x_k^{(j)})}-1}\,
Z_G(\,dx_1)\dots Z_G(dx_k).
\end{eqnarray*}
Let us make the substitution $y_j=Nx_j$, $j=1,\dots,k$, in the last
formula, and let us rewrite it in a form resembling 
formula~(\ref{($6.5'$)}).
To this end, let us introduce the measures~$G_N$ defined 
in~(\ref{(8.2)}). By Lemma~4.6 we can write
$$
Z_n^N\stackrel{\Delta}{=}\int f_N(y_1,\dots,y_k)
\tilde\chi_n(y_1+\cdots+y_k)\,
Z_{G_N}(\,dy_1)\dots Z_{G_N}(dy_k)
$$
with
\begin{equation}
f_N(y_1,\dots,y_k)=\prod_{j=1}^\nu
\frac{i(y_1^{(j)}+\cdots+y_k^{(j)})}
{\left(\exp\left\{i\frac1N(y_1^{(j)}
+\cdots+y_k^{(j)})\right\}-1\right)N},
\label{(8.5)}
\end{equation}
where $\tilde\chi_n(\cdot)$ is the Fourier transform of the 
indicator function of the unit cube 
$\prod\limits_{j=1}^\nu[n^{(j)},n^{(j)}+1)$.
(It follows from Lemma~8B formulated below and the Fubini 
theorem that the set, where the denominator of the function 
$f_N$ disappears, i.e. the set where 
$y_1^{(j)}+\cdots+y_k^{(j)}=2lN\pi$ with some integer
$l\neq0$ and $1\le j\le\nu$ has zero $G_N\times\cdots\times G_N$ 
measure. This means that the functions $f_N$ are well defined.)
The functions~$f_N$ tend to 1 uniformly in all bounded regions, 
and the measures~$G_N$ tend vaguely to~$G_0$ as $N\to\infty$ 
by Lemma~8.1. These relations suggest the following limiting 
procedure. The limit of $Z_n^N$ can be obtained by 
substituting $f_N$ with~1 and $G_N$ with~$G_0$ in the 
Wiener--It\^o integral expressing~$Z_n^N$. We want to justify
this formal limiting procedure. For this we have to show that 
the Wiener--It\^o integral expressing~$Z_n^N$ is essentially 
concentrated in a large bounded region independent of ~$N$. 
The $L_2$-isomorphism of Wiener--It\^o integrals can help us 
in showing that. We shall formulate a result in Lemma~8.3 
which is a useful tool for the justification of  the above 
limiting procedure.

Before formulating this lemma we make a small digression. It 
was explained that Wiener--It\^o integrals can be defined 
also with respect to random stationary fields $Z_G$ adapted 
to a stationary Gaussian random field whose spectral 
measure~$G$ may have atoms, and we can work with them 
similarly as in the case of non-atomic spectral measures. 
Here a lemma will be proved which shows that in the proof 
of Theorem~8.2 we do not need this observation, because
if the correlation function of the random field 
satisfies~(\ref{(8.1)}), then its spectral measure is 
non-atomic. Moreover, the measure~$G$ has an additional 
property which guarantees that the function 
$f_N(y_1,\dots,y_n)$ introduced in~(\ref{(8.5)}) can be 
defined in the space~$R^{k\nu}$ with the product measure 
$G_N\times\cdots\times G_N$.

\medskip\noindent
{\bf Lemma~8B.} {\it Let the correlation function of a 
stationary random field $X_n$, $n\in\textrm{\BBB Z}_\nu$, 
satisfy the relation $r(n)\le A|n|^{-\alpha}$ with some 
$A>0$ and $\alpha>0$ for all $n\in\textrm{\BBB Z}_\nu$, 
$n\neq0$. Then its spectral measure $G$ is non-atomic. 
Moreover, the hyperplanes $x^{(j)}=t$ have zero $G$ measure
for all $1\le j\le \nu$ and $t\in R^1$.}

\medskip\noindent
{\it Proof of Lemma 8B.} Lemma 8B clearly holds if $\alpha>\nu$,
because in this case the spectral measure~$G$ has even a density
 function $g(x)=\sum\limits_{n\in\textrm{\BBB Z}_\nu}e^{-i(n,x)}r(n)$.
On the other hand, the $p$-fold convolution of the spectral measure
$G$ with itself (on the torus $R^\nu/2\pi\textrm{\BBB Z}_\nu$) 
has Fourier transform, $r(n)^p$, $n\in{\BBB Z}^\nu$, and as 
a consequence in the case $p>\frac\nu\alpha$ this measure is 
non-atomic. Hence it is enough to show that if the convolution 
$G*G$ is a non-atomic measure, then the measure~$G$ is also
non-atomic. But this is obvious, because if there were a point 
$x\in R^\nu/2\pi\textrm{\BBB Z}_\nu$ such that $G(\{x\})>0$, 
then the relation $G*G(\{x+x\})>0$ would hold, and this is 
a contradiction. (Here addition is taken on the torus.) The 
second statement of the lemma can be proved with some small 
modifications of the previous proof, by reducing it to the 
one-dimensional case. \hfill$\qed$

\medskip
Now we formulate a result that helps us in carrying out some 
limiting procedures.

\medskip\noindent
{\bf Lemma~8.3.} {\it Let $G_N$, $N=1,2,\dots$, be a sequence
of non-atomic spectral measures on $R^\nu$ tending vaguely 
to a non-atomic spectral measure~$G_0$. Let a sequence of 
measurable functions $K_N=K_N(x_1,\dots,x_k)$, 
$N=0,1,2,\dots$, be given such that 
$K_N\in\bar{{\cal H}}_{G_N}^k$ for $N=1,2,\dots$. Assume 
further that these functions satisfy the following 
properties: For all $\varepsilon>0$ there exist some 
constants $A=A(\varepsilon)>0$ and $N_0=N_0(\varepsilon)>0$ 
and finitely many rectangles $P_1,\dots,P_M$ with some 
cardinality  $M=M(\varepsilon)$ on $R^{k\nu}$ such that
the following conditions~(a) and~(b) formulated below with 
the help of these numbers and rectangles are satisfied. (We 
call a set $P\in{\cal B}^{k\nu}$ a rectangle if it can be 
written in the form $P=L_1\times\cdots\times L_k$ with 
some bounded open sets $L_s\in{\cal B}^\nu$, $1\le s\le k$, 
with boundaries $\partial L_s$ of zero $G_0$~measure,
 i.e.\ $G_0(\partial L_s)=0$ for all $1\le s\le k$.)

\medskip
\begin{description}
\item[(a)] The function $K_0$ is continuous on the set
$B=[-A,A]^{k\nu}\setminus\bigcup\limits_{j=1}^MP_j$, and 
$K_N\to K_0$ uniformly on the set $B$ as $N\to\infty$. 
Besides, the hyperplanes $x_p=\pm A$ have zero 
$G_0$~measure for all $1\le p\le\nu$.
\item[(b)] $\int_{R^{k\nu}\setminus B}|K_N(x_1,\dots,x_k)|^2
G_N(\,dx_1)\dots G_N(dx_k)<\frac{\varepsilon^3}{k!}$ 
if $N=0$ or $N\ge N_0$, and 
$K_0(-x_1,\dots,-x_k)=\overline{K_0(x_1,\dots,x_k)}$ 
for all $(x_1,\dots,x_k)\in R^{k\nu}$.
\end{description}

\medskip
Then $K_0\in\bar{{\cal H}}_{G_0}^k$, and
$$
\int K_N(x_1,\dots,x_k)\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k) 
\stackrel{{\cal D}}{\rightarrow}
\int K_0(x_1,\dots,x_k)\,Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)
$$
%\begin{eqnarray*}
%&&\int K_N(x_1,\dots,x_k)\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k) \\
%&&\qquad \stackrel{{\cal D}}{\rightarrow}
%\int K_0(x_1,\dots,x_k)\,Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)
%\end{eqnarray*}
as $N\to\infty$, where $\stackrel{{\cal D}}{\rightarrow}$ denotes
convergence in distribution.}

\medskip\noindent
{\it Remark.}\/ In the proof of Theorem~8.2 or of its generalization
Theorem~$8.2'$ formulated later a simpler version of Lemma~8.3 with a
simpler proof would suffice. We could work with such a version where
the rectangles~$P_j$ do not appear. We formulated this somewhat more
complicated result, because it can be applied in the proof of more
general theorems, where the limit is given by such a Wiener--It\^o
integral whose kernel function may have discontinuities. Thus it
seemed to be better to present such a result even if its proof is
more complicated. The proof applies some arguments of Lemma~4.1.
To work out the details it turned out to be useful to introduce 
some metric in the space of probability measures which metricizes 
weak convergence. Although this may look a bit too technical, it 
made possible to carry out some arguments in a natural way. We
can tell with the help of this notion when two probability
measures are close to each other.

\medskip\noindent
{\it Proof of Lemma~8.3.}\/ 
Conditions~(a) and~(b) obviously imply that
$$
\int|K_0(x_1,\dots,x_k)|^2\,G_0(\,dx_1)\dots G_0(\,dx_k)<\infty,
$$
hence $K_0\in\bar{{\cal H}}_{G_0}^k$. Let us fix an 
$\varepsilon>0$, and let us choose some $A>0$, $N_0>0$ and 
rectangles $P_1,\dots,P_M$ which satisfy conditions~(a) 
and~(b) with this~$\varepsilon$. Then
\begin{eqnarray}
&&E\left[\int [1-\chi_B(x_1,\dots,x_k)]K_N(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)\right]^2 \nonumber \\
&&\qquad \le k!\int_{R^{k\nu}\setminus B}|K_N(x_1,\dots,x_k)|^2
G_N(\,dx_1)\dots G_N(\,dx_k)<\varepsilon^3
\label{(8.6)}
\end{eqnarray}
for $N=0$ or $N>N_0$, where $\chi_B$ denotes the indicator function
of the set~$B$ introduced in the formulation of condition~(a).

Since $B\subset [-A,A]^{k\nu}$, and $G_N\stackrel{v}{\rightarrow} G_0$,
hence $G_N\times\cdots\times G_N(B)<C(A)$ with an appropriate constant
$C(A)<\infty$ for all $N=0,1,\dots$.  Because of this estimate and
the uniform convergence $K_N\to K_0$ on the set~$B$ we have
\begin{eqnarray}
&&E\left[\int(K_N(x_1,\dots,x_k)-K_0(x_1,\dots,x_k))
\chi_B(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)\right]^2 \nonumber  \\
&&\qquad\le k!\int_B
|K_N(x_1,\dots,x_k)-K_0(x_1,\dots,x_k)|^2\,
G_N(\,dx_1)\dots G_N(\,dx_k)<\varepsilon^3 \label{(8.7)}
\end{eqnarray}
for $N>N_1$ with some $N_1=N_1(A,\varepsilon)$.

First we reduce the proof of Lemma~8.3 to the proof of 
the relation 
\begin{eqnarray}
&&\int K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k) \nonumber \\
&&\qquad \stackrel{{\cal D}}{\rightarrow}
\int K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)\,
Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)
\label{(8.8)}
\end{eqnarray}
with the help of formulas~(\ref{(8.6)}) and~(\ref{(8.7)}),
and then we shall prove~(\ref{(8.8)}). It is simpler
to carry out this reduction with the help of some metric on
the space of probability measure which induces weak 
convergence in this space. Hence I recall some classical 
notions and results about convergence of probability 
measures on a metric space which will be useful in our
considerations.

\medskip\noindent
{\bf Definition of Prokhorov Metric, and Its Properties.}
{\it Given a separable metric space $(X,\rho)$ with some 
metric~$\rho$ let ${\cal S}$ denote the space of probability 
measures on it. The  Prokhorov metric $\rho_P$ is the 
metric in the space  ${\cal S}$ defined by the formula
$\rho_P(\mu,\nu)=\inf\{\varepsilon\colon\; 
\mu(A)\le\nu(A^\varepsilon)+\varepsilon\textrm{ for all }
A\in{\cal A}\}$ for two probability measures 
$\mu,\nu\in{\cal S}$, where
$A^\varepsilon=\{x\colon\;\rho(x,A)<\varepsilon\}$. 
The above defined $\rho_P$ is really a metric on~${\cal S}$ 
(in particular, $\rho_P(\mu,\nu)=\rho_P(\nu,\mu)$)
which metricizes the weak convergence of probability 
measures in the metric space~$(X,\rho)$, i.e. 
$\mu_N\stackrel{w}{\rightarrow}\mu_0$ for a sequence of 
probability measures $N=0,1,2,\dots$ if and only if 
$\lim\limits_{N\to\infty}\rho_P(\mu_N,\mu_0)=0$.} 
\index{Prokhorov metric in the space of probability measures}

\medskip
The results formulated in this definition can be found e.g. 
in~\cite{r12b}. 
%R.M.~Dudley Distances of probability measures and random variables. 
%Ann.~Math.~Statist.~39, 1563--1572 (1968)). 
Let us also recall the definition of weak converges of probability 
measures on a metric space.

\medskip\noindent
{\bf Definition of Weak Convergence of Probability Measures on a
Metric Space.} {\it A sequence of probability measures $\mu_n$, 
$n=1,2,\dots$, on a metric space  $(X,\rho)$  converges weakly 
to a probability measure $\mu$  on this space, (in notation 
$\mu_n\stackrel{w}{\rightarrow}\mu$) if 
$\lim\limits_{n\to\infty}\int f(x)\mu_n(\,dx)\to\int f(x)\mu(\,dx)$
for all continuous and bounded functions on the space $(X,\rho)$.}
\index{weak convergence of probability measures}

\medskip
I formulated the above result for probability measures 
in a general metric space, but I shall work on the real 
line. Given a random variable~$\xi$ let $\mu(\xi)$ denote 
its distribution. Let us remark that the convergence 
$\xi_N\stackrel{{\cal D}}{\rightarrow}\xi_0$ as $N\to\infty$ 
of a sequence of random variables, $\xi_0,\xi_1,\xi_2,\dots$ 
is equivalent  to the statement 
$\mu(\xi_N)\stackrel{w}{\rightarrow}\mu(\xi_0)$ or
$\rho_P(\mu(\xi_N),\mu(\xi_0))\to0$ as $N\to\infty$. 
Hence by putting 
$\xi_N=k!I_{G_N}(K_N(x_1,\dots,x_k))$, $N=0,1,2,\dots$ 
we can reformulate the statement of Lemma~8.3 in the 
following way. For all $\varepsilon>0$ there exists 
some index $N'_0=N'_0(\varepsilon)$ such that  
$\rho_P(\mu(\xi_N),\mu(\xi_0))\le4\varepsilon$
for all~$N\ge N'_0$.

To reduce the proof of Lemma~8.3 to that of 
formula~(\ref{(8.8)}) first we show that for three 
random variables $\xi$, $\bar\xi$ and~$\eta$ such that
$P(|\eta|\ge\varepsilon)\le\varepsilon$ the inequality
\begin{equation}
\rho_P(\mu(\xi+\eta),\mu(\bar\xi))
\le\rho_P(\mu(\xi),\mu(\bar\xi))+\varepsilon \label{(8.9)}
\end{equation}
holds. 

As $\rho_P$ is a metric we can write 
$\rho_P(\mu(\xi+\eta),\mu(\bar\xi))\le
\rho_P(\mu(\xi+\eta),\mu(\xi))+
\rho_P(\mu(\xi),\mu(\bar\xi))$,
and to prove~(\ref{(8.9)}) it is enough to show that
$\rho_P(\mu(\xi+\eta),\mu(\xi))\le\varepsilon$ if
$P(|\eta|\ge\varepsilon)\le\varepsilon$.

This inequality holds, since 
$\{\omega\colon\;\xi(\omega)\in A\}
\subset\{\omega\colon\;\xi(\omega)+\eta(\omega)
\in A^\varepsilon\}\cup\{\omega\colon\;|\eta(\omega)|
\ge\varepsilon\}$, and as a consequence
$P(\xi\in A)\le P(\xi+\eta\in A^\varepsilon) 
+P(|\eta|\ge\varepsilon)
\le P(\xi+\eta\in A^\varepsilon)+\varepsilon$ 
for any set~$A\in{\cal B}_1$ if 
$P(|\eta|\ge\varepsilon)\le\varepsilon$. 
By the definition of the Prokhorov metric this means
that the desired inequality holds.


%Besides, $P(\xi\in A^\varepsilon)
%\le P(\bar\xi\in A^{\varepsilon+\delta})+\delta$ for all
%$\delta>\rho_P(\mu(\xi),\mu(\bar\xi))$. Hence
%$P(\xi+\eta\in A)\le 
%P(\bar\xi\in A^{\varepsilon+\delta})+\varepsilon+\delta$ for all
%$A\in{\cal B}_1$ and $\delta>\rho_P(\mu(\xi),\mu(\bar\xi))$, 
%i.e.\ $\rho_P(\mu(\xi+\eta),\mu(\bar\xi))\le\varepsilon+\delta$.
%This implies that the inequality 
%$\rho_P(\mu(\xi+\eta),\mu(\bar\xi))
%\le\rho_P(\mu(\xi),\mu(\bar\xi))+\varepsilon$ holds.

Put
\begin{eqnarray*}
\xi_N^{(1)}&=&k!I_{G_N}(K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)),\\
\xi_N^{(2)}&=&k!I_{G_N}(K_N(x_1,\dots,x_k)-K_0(x_1,\dots,x_k))
\chi_B(x_1,\dots,x_k)),\\
\xi_N^{(3)}&=&k!I_{G_N}(1-\chi_B(x_1,\dots,x_k))K_N(x_1,\dots,x_k)),\\
\xi_N&=&k! I_{G_N}(K_N)
\end{eqnarray*}
for all $N=0,1,2,\dots$. With this notation it follows from
relation~(\ref{(8.8)}) and the fact that the Prokhorov metric 
metricizes the weak convergence that 
$$
\rho_P(\mu(\xi_N^{(1)}),\mu(\xi_0^{(1)}))\le\varepsilon 
\quad \textrm{if } N\ge N'_1(\varepsilon) 
$$
with some threshold index~$N'_1(\varepsilon)$. 
Formulas~(\ref{(8.6)}) and~(\ref{(8.7)})
together with the Chebishev inequality imply that
$P(|\xi_N^{(2)}|\ge\varepsilon)\le\varepsilon$ and 
$P(|\xi_N^{(3)}|\ge\varepsilon)\le\varepsilon$
if $N\ge N'_2(\varepsilon)$ or $N=0$ with some threshold 
index~$N'_2(\varepsilon)$.
Besides, we have $\xi_0=\xi_0^{(1)}+\xi_0^{(3)}$ and
$\xi_N=\xi_N^{(1)}+\xi_N^{(2)}+\xi_N^{(3)}$ for $N=1,2,\dots$.
The above mentioned properties of the random variables we considered
together with relation~(\ref{(8.9)}) imply that
\begin{eqnarray*}
\rho_P(\mu(\xi_N),\mu(\xi_0))
&=&\rho_P(\mu(\xi_N^{(1)}+\xi_N^{(2)}+\xi_N^{(3)}),
\mu(\xi_0^{(1)}+\xi_0^{(3)}))\\
&\le&\rho_P(\mu(\xi_N^{(1)}+\xi_N^{(2)}+\xi_N^{(3)}),
\mu(\xi_0^{(1)}))+\varepsilon \\
&\le&\rho_P(\mu(\xi_N^{(1)}+\xi_N^{(2)}),\mu(\xi_0^{(1)}))
+2\varepsilon\\
&\le&\rho_P(\mu(\xi_N^{(1)}),\mu(\xi_0^{(1)}))+3\varepsilon
\le 4\varepsilon
\end{eqnarray*}
if $N\ge N'_0(\varepsilon)=\max(N'_1(\varepsilon),N'_2(\varepsilon))$. 
Hence Lemma~8.3 follows from~(\ref{(8.8)}).

\medskip
To prove~(\ref{(8.8)}) we will show that
$K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)$ can be well 
approximated by simple functions from 
$\hat{\bar{{\cal H}}}_{G_0}^k$ in the following way.
For all $\varepsilon'>0$ there exists a simple function 
$f_{\varepsilon'}\in\hat{\bar{{\cal H}}}^k_{G_0}$ such that
\begin{equation}
E\int( K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)
-f_{\varepsilon'}(x_1,\dots,x_k))^2
G_0(\,dx_1)\dots G_0(\,dx_k)\le\frac{{\varepsilon'}^3}{k!}
\label{(8.10)}
\end{equation}
and also
\begin{equation}
E\int( K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)
-f_{\varepsilon'}(x_1,\dots,x_k))^2
G_N(\,dx_1)\dots G_N(\,dx_k)\le\frac{{\varepsilon'}^3}{k!}
\label{(8.11)}
\end{equation}
if $N\ge N_0$ with some threshold index
$N_0=N_0(\varepsilon',K_0(\cdot)\chi_B(\cdot))$. Moreover, 
this simple  function $f_{\varepsilon'}$ can be chosen in 
such a way that it is adapted to such a regular system 
${\cal D}=\{\Delta_j,\;j=\pm1,\dots,\pm M\}$ whose 
elements have boundaries with zero $G_0$ measure, i.e. 
$G_0(\partial\Delta_j)=0$ for all $1\le |j|\le M$.

To prove~(\ref{(8.8)}) with the help of these 
estimates first we show that this function 
$f_{\varepsilon'}\in\hat{\bar{{\cal H}}}_{G_0}^k$
satisfies the relation
\begin{equation}
\int f_{\varepsilon'}(x_1,\dots,x_k)
\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)
\stackrel{{\cal D}}{\rightarrow}
\int f_{\varepsilon'}(x_1,\dots,x_k)
\,Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)
\label{(8.12)}
\end{equation}
%\begin{eqnarray}
%&&\int f_\varepsilon(x_1,\dots,x_k)
%\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)
%\nonumber \\
%&&\stackrel{{\cal D}}{\rightarrow}
%\int f_\varepsilon(x_1,\dots,x_k)
%\,Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)
%\label{(8.12)}
%\end{eqnarray}
as $N\to\infty$. In the proof of~(\ref{(8.12)}) we exploit
that we can take such a regular system 
${\cal D}=\{\Delta_j,\;j=\pm1,\dots,\pm M\}$ to which the 
function $f_{\varepsilon'}\in\hat{\bar{{\cal H}}}_{G_0}^k$ 
is adapted and which has the property $G_0(\partial\Delta_j)=0$ 
for all $j=\pm1,\dots,\pm M$. Besides, the spectral measures 
$G_N$ are such that $G_N\stackrel{v}{\rightarrow}G_0$. 
Hence the (Gaussian) random vectors 
$(Z_{G_N}(\Delta_j),\;j=\pm1,\dots,\pm M)$ converge in
 distribution to the (Gaussian) random vector
$(Z_{G_0}(\Delta_j),\;j=\pm1,\dots,\pm M)$ as $N\to\infty$.
The same can be told about such random variables that we 
get by putting the arguments of these random vectors to a 
continuous function (of $2M$ variables). Since the integrals 
in~(\ref{(8.12)}) are polynomials of these random vectors, 
we can apply these results for them, and they imply 
relation~(\ref{(8.12)}).

Put 
\begin{equation}
K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)
-f_{\varepsilon'}(x_1,\dots,x_k)=h_0(x_1,\dots,x_k). \label{(8.13)}
\end{equation}
By relations~(\ref{(8.10)}), (\ref{(8.11)}) and the Chebishev 
inequality
$P(|k!I_{G_0}(h_0)|\ge\varepsilon')\le\varepsilon'$
and $P(|k!I_{G_N}(h_0)\ge\varepsilon')\le\varepsilon'$
if $N\ge N_0$. Since
$I_{G_N}(K_N(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k))=
I_{G_N}(f_{\varepsilon'}(x_1,\dots,x_k)+(h_0(x_1,\dots,x_k))$, 
$N=0,1,2,\dots$, 
the above relations together with formulas~(\ref{(8.12)}) 
and~(\ref{(8.9)}) (with the number $\varepsilon'$ instead of 
$\varepsilon$) imply that 
\begin{eqnarray*}
&&\limsup_{N\to\infty}\rho_P(\mu(k!I_{G_N}(K_0(\cdot)\chi_B(\cdot))),
\mu(k!I_{G_0}(K_0(\cdot)\chi_B(\cdot)))) \\ 
&&=\limsup_{N\to\infty}
\rho_P(\mu(k!I_{G_N}(f_{\varepsilon'}(\cdot)+h_0(\cdot))),
\mu(k!I_{G_0}(f_{\varepsilon'}(\cdot)+h_0(\cdot)))) \\
&&=\limsup_{N\to\infty}
\rho_P(\mu(k!I_{G_N}(f_{\varepsilon'}(\cdot))+k!I_{G_N}(h_0(\cdot))),
\mu(k!I_{G_0}(f_{\varepsilon'}(\cdot))+k!I_{G_0}(h_0(\cdot)))) \\
&&\qquad \le\limsup_{N\to\infty}
\rho_P(\mu(k!I_{G_N}(f_{\varepsilon'}(\cdot))),
\mu(k!I_{G_0}(f_{\varepsilon'}(\cdot))))+2\varepsilon'=2\varepsilon'.
\end{eqnarray*}
Since this inequality holds for all $\varepsilon'>0$ this
implies relation~(\ref{(8.8)}). To complete the proof of
Lemma~8.3 we have to justify relations~(\ref{(8.10)}) 
and~(\ref{(8.11)}).

\medskip
Relation~(\ref{(8.10)}) is actually a version 
of Lemma~4.1, but it states a slightly stronger 
approximation result under the conditions of 
Lemma~8.3. The statement that for all $\varepsilon'$ 
the function $K_0(\cdot)\chi_B(\cdot)$ can be 
approximated with a simple function  
$f_{\varepsilon'}(x_1,\dots,x_k)$ which
satisfies~(\ref{(8.10)}) agrees with Lemma~4.1. 
But now we want to choose such a simple function 
$f_{\varepsilon'}$ which is adapted to a regular 
system ${\cal D}=\{\Delta_j,\;j=\pm1,\dots,\pm M\}$ 
with such elements that have the additional property 
$G_0(\partial \Delta_j)=0$  for all indices~$j$. 
A function $f_{\varepsilon'}$ with these
properties can be constructed by means of a slight 
modification of the proof of Lemma~4.1. We exploit 
that in the present case the function 
$K_0(\cdot)\chi_B(\cdot)$ is almost everywhere 
continuous with respect to the product measure 
$G_0^k=\underbrace{G_0\times\cdots\times G_0}_{k\textrm{ times}}$.
This property is needed in the first step of the 
construction, where we reduce the approximation result
we want to prove to a slightly modified version of 
{\it Statement~A}. 

In this modified version of {\it Statement~A}\/ we 
want to find a good approximation of the indicator
function of such sets $A$ which satisfies not only 
the properties demanded in {\it Statement~A},\/ but 
also the identities $G_0(\partial A)=0$ and 
$G_0(\partial A_1)=0$ hold for them. On the other hand, 
we demand the identity $G_0(\partial B)=0$ also for the 
set $B$ whose indicator function is the approximating 
function in {\it Statement A}. To carry out the 
reduction, needed in this case we approximate the 
function~$K_0(\cdot)\chi_B(\cdot)$ with such an 
elementary function (a function taking finitely many 
values) whose level sets have boundaries with zero 
$G_0^k=G_0\times\dots\times G_0$ measure. This is possible, 
since the boundaries of these level sets consist of such 
points where either the function $K_0(\cdot)\chi_B(\cdot)$ 
takes the value from an appropriately chosen finite set, or 
this function is discontinuous. At this point we exploit 
that the function $K_0(\cdot)\chi_B(\cdot)$ is almost 
everywhere continuous with respect to the measure~$G_0$. 

To complete the reduction of the proof of~(\ref{(8.10)}) 
to the new version of {\it Statement~A}\/ we still have 
to show that if the set $A$ can be written in the form  
$A=A_1\cup(-A_1)$ such that $A_1\cap(-A_1)=\emptyset$, 
and $G_0^k(\partial A_1)=0$, then for all $\eta>0$ there 
is some $\bar A_1=\bar A_1(\eta)\subset A_1$ such that 
$G_0^k(A\setminus(\bar A_1\cup(-\bar A_1))\le\eta$, 
$\rho(\bar A_1,-\bar A_1)>0$, and 
$G_0^k(\partial \bar A_1)=0$. Indeed, there is a
compact set $K\subset A_1$ such that 
$G^k(A_1\subset K)\le\frac\eta2$. Then also
the relation $\rho(K,-K)=\delta>0$ holds. By the
Heine--Borel theorem we can find an open set $G$
such that $K\subset G\subset K^{\delta/3}$ with
$K^{\delta/3}=\{x\colon \rho(x,K)<\frac\delta3\}$,
and $G_0^k(\partial G)=0$. Then the set 
$\bar A_1=A_1\cap G$ satisfies the desired
properties.

After making the reduction of the result we want to
prove to this modified version of {\it Statement A}\/
we can follow the construction of 
Lemma~4.1, but we choose in each step sets with zero 
$G_0\times\cdots\times G_0$ boundary.

A more careful analysis shows that the function constructed
in such a way satisfies also~(\ref{(8.11)}) for $N\ge N_0$ 
with a sufficiently large threshold index~$N_0$. Here we 
exploit that $G_N\stackrel{v}{\rightarrow}G_0$. This 
enables us to show that the estimates we need in the 
construction hold not only with respect to the spectral 
measure~$G_0$ but also with respect to the spectral 
measures~$G_N$ with a sufficiently large index~$N$. 
We can get another explanation of the estimate~(\ref{(8.11)}) 
by exploiting that the function $h_0(x_1,\dots,x_k)$ defined 
in~(\ref{(8.13)}) is almost everywhere continuous with  
respect to the measure $G_0\times\cdots\times G_0$. It can
be shown that the vague convergence has similar properties 
as the weak convergence. In particular, the above mentioned 
almost everywhere continuity implies that
$$
\lim_{N\to\infty}\int h_0(x_1,\dots,x_k)G_N(\,dx_1)\dots G_N(\,dx_k)
=\int h_0(x_1,\dots,x_k)G_0(\,dx_1)\dots G_0(\,dx_k).
$$
\hfill$\qed$

\medskip\noindent
{\it Remark.} In Lemma~8.3 we proved the convergence of 
Wiener--It\^o integrals with respect to random spectral 
measures $Z_{G_N}$ corresponding to spectral measures 
$G_N$ on the Euclidean space $R^\nu$ under appropriate 
conditions. There is a natural version of this result  
which we get by considering Wiener--It\^o integrals 
$k!I_{G_N}(K_N)$ on the torus of size $2C_N\pi$ with 
some numbers $C_N\to\infty$ as $N\to\infty$. To find
a good formulation of the result in this case observe
that the torus $R^\nu/2\pi\textrm{\BBB Z}_\nu$ can be
identified with the set
$[-C_N\pi,C_N\pi)^\nu\subset R^\nu$ in a natural way.
This identification enables us to consider the spectral
measure $G_N$ as a measure on $[-C_N\pi,C_N\pi)^\nu$ 
and the function $K_N$ as a function on this set, 
which can be extended to a function on $R^\nu$, 
periodic in all of its coordinates with periodicity 
$2\pi C_N$. With such a notation we demand in this 
version of Lemma~8.3 that 
$G_N\stackrel{v}{\rightarrow}G_0$, and conditions~(a)
and~(b) hold with these (non-atomic) measures $G_N$ and
functions~$K_N$. This version of Lemma~8.3 can be proved
in almost the same way. We can reduce its proof to the
verification of formula~(\ref{(8.8)}), and after this it
has no importance whether we work in~$R^\nu$ or 
in~$[-C_N\pi,C_N\pi)^\nu$.


 
\medskip
Now we turn to the proof of Theorem~8.2.

\medskip\noindent
{\it Proof of Theorem~8.2.} We want to prove that for all
positive integers $p$, real numbers $c_1,\dots,c_p$  and
$n_l\in\textrm{\BBB Z}_\nu$, $l=1,\dots,p$,
$$
\sum_{l=1}^p c_l Z^N_{n_l}\stackrel{{\cal D}}{\rightarrow}
\sum_{l=1}^p c_lZ^*_{n_l},
$$
since this relation also implies the convergence of the
multi-dimensional distributions. Applying the same calculation
as before we get with the help of Lemma~4.6 that
$$
\sum_{l=1}^p c_lZ_{n_l}^N=\frac1{A_N}\sum_{l=1}^p c_l\int
\sum_{j\in B_{n_l}^N}e^{i(j,x_1+\cdots+x_k)}
\,Z_G(\,dx_1)\dots Z_G(\,dx_k),
$$
and
$$
\sum_{l=1}^p c_lZ^N_{n_l}\stackrel{\Delta}{=}
\int K_N(x_1,\dots,x_k)
\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)
$$
with
\begin{eqnarray}
K_N(x_1,\dots,x_k)&=& \frac1{N^\nu}\sum_{l=1}^p c_l\sum_{j\in B_{n_l}^N}
\exp\left\{i\left(\frac jN,x_1+\cdots+x_k\right)\right\} \nonumber  \\
&=&f_N(x_1,\dots,x_k)\sum_{l=1}^p c_l\tilde\chi_{n_l}(x_1+\cdots+x_k).
\label{(8.14)}
\end{eqnarray}
with the function~$f_N$ defined in~(\ref{(8.5)}) and 
the measure~$G_N$ defined in~(\ref{(8.2)}), The function  
$\tilde\chi_n(\cdot)$ denotes again the Fourier 
transform of the indicator function of the unit cube 
$\prod\limits_{j=1}^\nu[n^{(j)},n^{(j)}+1)$, 
$n=(n^{(1)},\dots n^{(\nu)})$.

Let us define the function
$$
K_0(x_1,\dots,x_k)=\sum_{l=1}^p c_l\tilde\chi_{n_l}(x_1+\cdots+x_k)
$$
and the measures~$\mu_N$ on $R^{k\nu}$ by the formula
\begin{eqnarray}
&&\mu_N(A)=\int_A|K_N(x_1,\dots,x_k)|^2 G_N(\,dx_1)\dots G_N(\,dx_k),
\nonumber  \\
&&\qquad \quad A\in{\cal B}^{k\nu} \textrm{ and } N=0,1,\dots.
\label{(8.15)}
\end{eqnarray}
In the case $N=0$ $G_0$ is the vague limit of the measures~$G_N$.

We prove Theorem~8.2 by showing that Lemma~8.3 can be 
applied with these spectral measures~$G_N$ and
functions~$K_N$. (We choose no exceptional rectangles~$P_j$ 
in this application of Lemma~8.3.) Since 
$G_N\stackrel{v}{\rightarrow} G_0$, and $K_N\to K_0$ 
uniformly in all bounded regions in $R^{k\nu}$, it is 
enough  to show, beside the proof of Lemma~8.1, that 
the measures $\mu_N$, $N=1,2,\dots$, tend weakly to the 
(necessarily finite) measure~$\mu_0$ which is also defined 
in~(\ref{(8.15)}), (in notation 
$\mu_N\stackrel{w}{\rightarrow}\mu_0$), i.e.
$\int f(x)\mu_N(\,dx)\to\int f(x)\mu_0(\,dx)$ for all 
continuous and bounded functions~$f$ on~$R^{k\nu}$. 
\index{weak convergence of probability measures}
Then this convergence implies condition~(b) in Lemma~8.3. 
Moreover, it is enough to show the slightly weaker statement 
by which there exists some finite measure $\bar\mu_0$ such 
that $\mu_N\stackrel{w}{\rightarrow}\bar\mu_0$, since then 
$\bar\mu_0$ must coincide with $\mu_0$ because of the 
relations $G_N\stackrel{v}{\rightarrow} G_0$ and 
$K_N\to K_0$ uniformly in all bounded regions of $R^{k\nu}$, 
and $K_0$ is a continuous function. This implies that
$\mu_N\stackrel{v}{\rightarrow}\mu_0$, and $\mu_0=\bar\mu_0$.

There is a well-known theorem in probability theory about the
equivalence between weak convergence of finite measures and the
convergence of their Fourier transforms. It would be natural to
apply this theorem for proving
$\mu_N\stackrel{w}{\rightarrow}\bar\mu_0$. On the other hand, we have
the additional information that the measures $\mu_N$, $N=1,2,\dots$,
are concentrated in the cubes $[-N\pi,N\pi)^{k\nu}$, since the
spectral measure~$G$ is concentrated in $[-\pi,\pi)^\nu$. It
is more fruitful to apply a version of the above mentioned theorem,
where we can exploit our additional information. We formulate the
following

\medskip\noindent
{\bf Lemma 8.4.} {\it Let $\mu_1,\mu_2,\dots$ be a sequence 
of finite measures on $R^l$ such that 
$\mu_N(R^l\setminus [-C_N\pi,C_N\pi)^l)=0$
for all $N=1,2,\dots$, with some sequence $C_N\to\infty$ as
$N\to\infty$. Define the modified Fourier transform
\index{modified Fourier transform}
$$
\varphi_N(t)=\int_{R^l} 
\exp\left\{i\left(\frac{[tC_N]}{C_N},x\right)\right\}
\mu_N(\,dx), \quad t\in R^l,
$$
where $[tC_N]$ is the integer part of the vector $tC_N\in R^l$. 
(For an $x\in R^l$ its integer part $[x]$ is the vector 
$n\in\textrm{\BBB Z}_l$ for which $x^{(p)}-1<n^{(p)}\le x^{(p)}$ 
if $x^{(p)}\ge0$, and $x^{(p)}\le n^{(p)}< x^{(p)}+1$ if 
$x^{(p)}<0$ 
for all $p=1,2,\dots,l$.) If for all $t\in R^l$ 
the sequence $\varphi_N(t)$ tends to a function $\varphi(t)$ 
continuous at the origin, then the measures $\mu_N$ weakly 
tend to a finite measure~$\mu_0$, and $\varphi(t)$ is the 
Fourier transform of~$\mu_0$.}

\medskip
I make some comments on the conditions of Lemma~8.4. Let us observe
that if the measures~$\mu_N$ or a part of them are shifted with a
vector $2\pi C_N u$ with some $u\in\textrm{\BBB Z}_l$, then their
modified Fourier transforms $\varphi_N(t)$ do not change because of
the periodicity of the trigonometrical functions $e^{i(j/C_N,x)}$,
$j\in\textrm{\BBB Z}_l$. On the other hand, these new measures
which are not concentrated in $[-C_N\pi,C_N\pi)^l$, have no limit.
Lemma~8.4 states that if the measures~$\mu_N$ are concentrated in the
cubes $[-C_N\pi,C_N\pi)^l$, then the convergence of their modified
Fourier transforms defined in Lemma~8.4, which is a weaker condition,
than the convergence of their Fourier transforms, also implies their
convergence to a limit measure.

\medskip\noindent
{\it Proof of Lemma~8.4.}\/ The proof is a natural modification of
the proof about the equivalence of weak convergence of measures and
the convergence of their Fourier transforms. First we show that
for all $\varepsilon>0$ there exits some $K=K(\varepsilon)$ such that
\begin{equation}
\mu_N(x\colon\; x\in R^l,\; |x^{(1)}|>K)<\varepsilon 
\quad \textrm{for all \ }N\ge1.
\label{(8.16)}
\end{equation}

As $\varphi(t)$ is continuous at the origin there is some $\delta>0$
such that
\begin{equation}
|\varphi(0,\dots,0)-\varphi(t,0,\dots,0)|<\frac\varepsilon2 
\quad\textrm{if \ } |t|<\delta. \label{(8.17)}
\end{equation}
We have
\begin{equation}
0\le \textrm{Re}\,[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]
\le2\varphi_N(0,\dots,0) \label{(8.18)}
\end{equation}
for all $N=1,2,\dots$. The sequence in the middle term 
of~(\ref{(8.18)}) tends to 
$$
\textrm{Re}\,[\varphi(0,\dots,0)-\varphi(t,0,\dots,0)]
$$ 
as $N\to\infty$. The right-hand side of~(\ref{(8.18)}) is a 
bounded sequence, since it is convergent. Hence the dominated 
convergence theorem can be applied for the functions
$\textrm{Re}\,[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]$.
Then we get because of the condition $C_N\to\infty$ and
relation~(\ref{(8.17)}) that
\begin{eqnarray*}
&&\lim_{N\to\infty} \int_0^{[\delta C_N]/C_N} \frac1\delta\,
\textrm{Re}\,[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]\,dt\\
&&\qquad=\int_0^\delta\frac1\delta\,
\textrm{Re}\,[\varphi(0,\dots,0)-\varphi(t,0,\dots,0)]\,dt
<\frac\varepsilon2
\end{eqnarray*}
with the number $\delta>0$ appearing in~(\ref{(8.17)}). Hence
\begin{eqnarray*}
\frac\varepsilon2&>& \lim_{N\to\infty} 
\int_0^{[\delta C_N]/C_N}\frac1\delta\,
\textrm{Re}\,
[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]\,dt \\
&=&\lim_{N\to\infty}\int
\left(\frac1\delta\int_0^{[\delta C_N]/C_N}
\textrm{Re}\, [1-e^{i[tC_N]x^{(1)}/C_N}]\,dt\right) 
\mu_N(\,dx)\\
&=&\lim_{N\to\infty}\int
\frac1{\delta C_N}\sum_{j=0}^{[\delta C_N]-1}
\textrm{Re}\,\left[1-e^{ijx^{(1)}/C_N}\right]\mu_N(\,dx)\\
&\ge&\limsup_{N\to\infty} \int_{\{|x^{(1)}|>K\}}
\frac1{\delta C_N} \sum_{j=0}^{[\delta C_N]-1} 
\textrm{Re}\, \left[1-e^{ijx^{(1)}/C_N}\right]\mu_N(\,dx)\\
&=&\limsup_{N\to\infty}\int_{\{|x^{(1)}|>K\}}
\left(1-\frac1{\delta C_N}
\textrm{Re}\,\frac{1-e^{i[\delta C_N]x^{(1)}/C_N}}
{1-e^{ix^{(1)}/C_N}}\right)\mu_N(\,dx)
\end{eqnarray*}
with an arbitrary $K>0$. (In the last but one step of this 
calculation we have exploited that 
$\frac1{\delta C_N}\sum\limits_{j=0}^{[\delta C_N]-1}
\textrm{Re}\,[1-e^{ijx^{(1)}/C_N}]\ge0$ for all 
$x^{(1)}\in R^1$.)

Since the measure $\mu_N$ is concentrated in
$\{x\colon\, x\in R^l,\;|x^{(1)}|\le C_N\pi\}$, and
\begin{eqnarray*}
\textrm{Re}\,\frac{1-e^{i[\delta C_N]x^{(1)}/C_N}}
{1-e^{ix^{(1)}/C_N}}
&=&\frac{\textrm{Re}\,\left(i e^{-ix^{(1)}/2C_N}
\left(1-e^{i[\delta C_N]x^{(1)}/C_N}\right)\right)}
{i(e^{-ix^{(1)}/2CN}-e^{ix^{(1)}/2C_N})}\\
&\le&\frac1{\left|\sin \left(\dfrac{x^{(1)}}{2C_N}\right)\right|}
\le \frac{C_N\pi}{|x^{(1)}|}
\end{eqnarray*}
if $|x^{(1)}|\le C_N\pi$, (here we exploit that
$|\sin u|\ge\frac2\pi|u|$ if $|u|\le\frac\pi2$), hence we have with
the choice $K=\frac{2\pi}{\delta}$
$$
\frac\varepsilon2>\limsup_{N\to\infty}\int_{\{|x^{(1)}|>K\}}
\left(1-\left|\frac\pi{\delta x^{(1)}}\right|\right)\mu_N(\,dx)
\ge\limsup_{N\to\infty}\frac12\mu_N(|x^{(1)}|>K).
$$
As the measures $\mu_N$ are finite the inequality
$\mu_N(|x^{(1)}|>K)<\varepsilon$ holds for each index~$N$ with a
constant~$K=K(N)$ that may depend on~$N$. Hence the above
inequality implies that formula~(\ref{(8.16)}) holds for all $N\ge1$
with a possibly larger index~$K$ that does not depend on~$N$.

Applying the same argument to the other coordinates we 
find that for all $\varepsilon>0$ there exists some 
$C(\varepsilon)<\infty$ such that
$$
\mu_N\left(R^l\setminus[-C(\varepsilon),
C(\varepsilon)]^l\right)<\varepsilon \quad 
\textrm{for all } N=1,2,\dots.
$$

Consider the usual Fourier transforms
$$
\tilde\varphi_N(t)=\int_{R^l}e^{i(t,x)}\mu_N(\,dx), \quad t\in R^l.
$$
Then
\begin{eqnarray*}
|\varphi_N(t)-\tilde\varphi_N(t)|&\le& 2\varepsilon
+\int_{[-C(\varepsilon),C(\varepsilon)]}
\left|e^{i(t,x)}-e^{i([tC_N]/C_N,x)}\right|\mu_N(\,dx)\\
&\le& 2\varepsilon
+\frac{lC(\varepsilon)}{C_N}\mu_N(R^l)
\end{eqnarray*}
for all $\varepsilon>0$. Hence $\tilde\varphi_N(t)-\varphi_N(t)\to0$ as
$N\to\infty$, and $\tilde\varphi_N(t)\to\varphi(t)$. (Observe that
$\mu_N(R^l)=\varphi_N(0)\to\varphi(0)<\infty$ as $N\to\infty$, hence
the measures~$\mu_N(R^l)$ are uniformly bounded, and $C_N\to\infty$
by the conditions of Lemma~8.4.) Then Lemma~8.4 follows from standard
theorems on Fourier transforms. \hfill$\qed$

\medskip
We return to the proof of Theorem~8.2. We apply Lemma~8.4 with 
$C_N=N$ and $l=k\nu$ for the measures $\mu_N$ defined 
in~(\ref{(8.15)}). Because of the middle term in~(\ref{(8.14)}) 
we can write the modified Fourier transform $\varphi_N$ of the
measure $\mu_N$ as
\begin{equation}
\varphi_N(t_1,\dots,t_k)=\sum_{r=1}^p\sum_{s=1}^p c_rc_s
\psi_N(t_1+n_r-n_s,\dots,t_k+n_r-n_s) \label{(8.19)}
\end{equation}
with
\begin{eqnarray}
&&\psi_N(t_1,\dots,t_r)=\frac1{N^{2\nu}}\int\exp\left\{
i\frac1N((j_1,x_1)+\cdots+(j_k,x_k))\right\}  \nonumber  \\
&&\qquad\qquad\sum_{u\in B^N_0}\sum_{v\in B^N_0}
\exp\left\{i\left(\frac{u-v}N,x_1+\cdots+x_k\right)\right\}
G_N(\,dx_1)\dots G_N(\,dx_k) \nonumber  \\
&&\qquad=\frac1{N^{2\nu-k\alpha}L(N)^k}\sum_{u\in B_0^N}\sum_{v\in B_0^N}
r(u-v+j_1)\cdots r(u-v+j_k), \label{(8.20)}
\end{eqnarray}
where  $j_p=[t_pN]$, $t_p\in R^\nu$, $p=1,\dots,k$. 

The asymptotical behaviour of $\psi_N(t_1,\dots,t_k)$ for 
$N\to\infty$ can be investigated by the help of the last 
relation and formula~(\ref{(8.1)}). Rewriting the last 
double sum in the form of a single sum  by fixing first 
the variable $l=u-v\in [-N,N]^\nu\cap\textrm{\BBB Z}_\nu$, 
and then summing up for~$l$ one gets
$$
\psi_N(t_1,\dots,t_k)=\int_{[-1,1]^\nu} f_N(t_1,\dots,t_k,x)\,dx
$$
with
\begin{eqnarray*}
&&f_N(t_1,\dots,t_k,x) \\
&&\qquad =\left(1-\frac{[|x^{(1)}N|]}N\right)\cdots
\left(1-\frac{[|x^{(\nu)}N|]}N\right)
\frac{r([xN]+j_1)}{N^{-\alpha}L(N)}\cdots
\frac{r([xN]+j_k)}{N^{-\alpha}L(N)}.
\end{eqnarray*}
(In the above calculation we exploited that in the last sum of
formula~(\ref{(8.20)}) the number of pairs $(u,v)$ for which
$u-v=l=(l_1,\dots,l_\nu)$ equals $(N-|l_1|)\cdots(N-|l_\nu|)$.)

Let us fix some vector $(t_1,\dots,t_k)\in R^{k\nu}$. It can 
be seen with the help of formula~(\ref{(8.1)}) that 
for all $\varepsilon>0$ the convergence
\begin{equation}
f_N(t_1,\dots,t_k,x)\to f_0(t_1,\dots,t_k,x) \label{(8.21)}
\end{equation}
holds uniformly with the limit function
\begin{equation}
f_0(t_1,\dots,t_k,x)=(1-|x^{(1)}|)\dots(1-|x^{(\nu)}|)
\frac{a\left(\frac{x+t_1}{|x+t_1|}\right)}{|x+t_1|^\alpha}\dots
\frac{a\left(\frac{x+t_k}{|x+t_k|}\right)}{|x+t_k|^\alpha}
\label{(8.22)}
\end{equation} 
on the set $x\in[-1,1]^\nu\setminus
\bigcup\limits_{p=1}^k\{x\colon\;|x+t_p|>\varepsilon\}$. 

We claim that
$$
\psi_N(t_1,\dots,t_k)\to\psi_0(t_1,\dots,t_k)
=\int_{[-1,1]^\nu}f_0(t_1,\dots,t_k,x)\,dx,
$$
and $\psi_0$ is a continuous function.

This relation implies that $\mu_N\stackrel{w}{\rightarrow}\mu_0$. To
prove it, it is enough to show beside formula~(\ref{(8.21)}) that
\begin{equation}
\left|\int_{|x+t_p|<\varepsilon} 
f_0(t_1,\dots,t_k,x)\,dx\right|<C(\varepsilon),
\quad p=1,\dots,k, \label{(8.23)}
\end{equation}
and
\begin{equation}
\int_{|x+t_p|<\varepsilon} |f_N(t_1,\dots,t_k,x)|\,dx<C(\varepsilon),
\quad p=1,\dots,k, \quad \textrm{and } N=1,2,\dots \label{($8.24'$)}
\end{equation}
with a constant $C(\varepsilon)$ such that 
$C(\varepsilon)\to0$ as $\varepsilon\to0$.

By formula~(\ref{(8.22)}) and H\"older's inequality
\begin{eqnarray*}
\left|\int_{|x+t_p|<\varepsilon} f_0(t_1,\dots,t_k,x)\,dx\right|
&&\le C\prod_{1\le l\le k,\,l\neq p}\left[\int_{x\in[-1,1]^\nu}
|x+t_l|^{-k\alpha}\,dx\right]^{1/k} \\
&&\qquad \left[\int_{|x+t_p|\le\varepsilon}|x
+t_p|^{-k\alpha}\,dx\right]^{1/k}
\le C'\varepsilon^{\nu/k-\alpha}
\end{eqnarray*}
with some appropriate $C>0$ and $C'>0$, since $\nu-k\alpha>0$,
and $a(\cdot)$ is a bounded function. Similarly,
\begin{eqnarray}
\int_{|x+t_p|<\varepsilon} |f_N(t_1,\dots,t_k,x)|\,dx
&&\le \prod_{1\le l\le k,\,l\neq p}\left[\int_{x\in[-1,1]^\nu}
\frac{|r([xN]+j_l)|^k}{N^{-k\alpha}L(N)^k}\,dx\right]^{1/k},
\nonumber \\
&&\qquad \left[\int_{|x+t_p|\le\varepsilon}
\frac{|r([xN]+j_p)|^k}{N^{-k\alpha}L(N)^k}\,dx\right]^{1/k}.
\label{(8.25)}
\end{eqnarray}

It is not difficult to see with the help of Karamata's theorem 
that if $L(t)$, $t\ge1$, is a slowly varying function which is 
bounded in all finite intervals, then for all numbers $\eta>0$ 
and $K>0$ there are some constants $K_1=K_1(\eta,K)>0$, and 
$C=C(\eta,K)>0$ together with a threshold index~$N_0=N_0(\eta,K)$ 
such that
$$
\frac{L(uN)}{L(N)}\le Cu^{-\eta} \quad\textrm{if } uN>K_1, \;\; u\le K,
\textrm{ \ and } N\ge N_0.
$$
Hence formula~(\ref{(8.1)}) implies that
\begin{eqnarray}
|r([xN]+[t_lN])=|r([xN]+j_l)|
&&\le CN^{-\alpha}L(N)|x+t_l|^{-\alpha-\eta} \nonumber \\ 
&&\qquad \textrm{if } |x+t_l|\le K \textrm{ and } N\ge N_0. 
\label{(8.26)}
\end{eqnarray}
Relation~ (\ref{(8.26)}) follows from the previous relation 
and~(ref{(8.1)}) if $|[xN]+[t_lN]|\ge K_1$. It also holds if
$|[xN]+[t_lN]|\le K_1$, since in this case the left-hand side
can be bounded by the inequality $|r([xN]+[t_lN]|\le1$, while 
the right-hand side of~(\ref{(8.26)}) is greater than 1 with 
the choice of a sufficiently large constant~$C$ (depending on 
$\eta$ and $K_1$). This follows from the relation
 $|x+t|^{-\alpha-\eta}
=N^{\alpha+\eta}|N(x+t)|^{-\alpha-\eta}\ge C_1N^{\alpha+\eta}$
if $|[xN]+[t_lN]|\le K_1$, and $L(N)\ge N^{-\eta}$.

We get with the help of~(\ref{(8.26)}) that
\begin{eqnarray*}
&&\int_{|x+t_p|<\varepsilon}\frac{|r([xN]+j_p)|^k}{N^{-k\alpha}L(N)^k}\,dx
\le B\int_{|x+t_p|<\varepsilon}|x+t_p|^{-k(\alpha+\eta)}\,dx
\le B'\varepsilon^{\nu-k(\alpha +\eta)}\\
&&\int_{x\in[-1,1]^\nu}\frac{|r([xN]+j_l)|^k}{N^{-k\alpha}L(N)^k}\,dx
\le B''.
\end{eqnarray*}
for a sufficiently small constant $\eta>0$ with some constants
$B,B',B''<\infty$ depending on $\eta$ and $t_p$, $1\le p\le k$.


Therefore we get from~(\ref{(8.25)}), by choosing an $\eta>0$ 
such that $k(\alpha+\eta)<\nu$, that the inequality
$$
\int_{|x+t_p|<\varepsilon} |f_N(t_1,\dots,t_k,x)|\,dx
\le C\varepsilon^{\nu/k-(\alpha +\eta)}
$$
holds with some $C<\infty$. The right-hand side of this 
inequality tends to zero as $\varepsilon\to0$. Hence 
we proved beside~(\ref{(8.21)}) formulae~(\ref{(8.23)})
and~(\ref{($8.24'$)}), and they have the consequence that
$\psi_N(t_1,\dots,t_k)\to\psi_0(t_1,\dots,t_k)$.
Since $\psi(t_1,\dots,t_k)$ is a continuous function
relation~(\ref{(8.19)}) with Lemma~8.4 imply that the
measures $\mu_N$ introduced in~(\ref{(8.18)}) converge
weakly to a probability measure as $N\to\infty$, and as we
saw at the beginning of the proof of Theorem~8.2 this
limit measure must be~$\mu_0$.

Hence we can apply Lemma~8.3 for the spectral measures~$G_N$ 
and functions $K_N(\cdot)$, $N=0,1,2,\dots$, defined in 
Theorem~8.2. In this application of Lemma~8.3 we choose no 
rectangles~$P_N$. The convergence 
$G_N\stackrel{v}{\rightarrow} G_0$ follows from Lemma~8.1. 
Conditions~(a) and~(b) also hold with the choice of a 
sufficiently large number~$A=A(\varepsilon)$. The hard 
point of the proof was the checking of condition~(b). 
This follows from the relation 
$\mu_N\stackrel{w}{\rightarrow} \mu_0$. Thus we have proved 
Theorem~8.2 with the help of Lemma~8.1. \hfill$\qed$

It remained to prove Lemma~8.1.

\medskip\noindent
{\it Proof of Lemma~8.1.}\/ Introduce the notation
$$
K_N(x)=\prod_{j=1}^\nu\frac {e^{ix^{(j)}}-1}{N(e^{ix^{(j)}/N}-1)},
\quad N=1,2,\dots,
$$
and
$$
K_0(x)=\prod_{j=1}^\nu\frac {e^{ix^{(j)}}-1}{ix^{(j)}}.
$$
Let us consider the measures $\mu_N$ defined in 
formula~(\ref{(8.15)}) in the special case $k=1$ with $p=1$, 
$c_1=1$ in the definition of the function $K_N(\cdot)$, i.e. put 
$$
\mu_N(A)=\int_A |K_N(x)|^2\,G_N(\,dx),
\quad A\in {\cal B}^\nu, \quad N=1,2,\dots.
$$
We have already seen in the proof of Theorem~8.2 that
$\mu_N\stackrel{w}{\rightarrow} \mu_0$ with some finite 
measure~$\mu_0$, and the Fourier transform of~$\mu_0$ is
$$
\varphi_0(t)=\int_{[-1,1]^\nu}(1-|x^{(1)}|)\cdots (1-|x^{(\nu)}|)
\frac{a\left(\frac{x+t}{|x+t|}\right)}{|x+t|^\alpha}\,dx.
$$
Moreover, since $|K_N(x)|^2\to |K_0(x)|^2$ uniformly in 
any bounded domain, it is natural to expect that 
$G_N\stackrel{v}{\rightarrow}G_0$ with 
$G_0(\,dx)=\frac1{|K_0(x)|^2}\mu_0(\,dx)$. But $K_0(x)=0$
in some points, and                                       %
the function $K_0(\cdot)^{-2}$ is not continuous in these 
points. As a consequence, we cannot give a direct proof 
of the above statement. Hence we apply instead a modified 
version of this method. First we prove the following 
result about the behaviour of the restrictions of the 
measures $G_N$ to appropriate cubes: 

For all $T\ge1$ there is a finite measure 
$G_0^T$ concentrated on $(-T\pi,T\pi)^\nu$ such that
\begin{equation}
\lim_{N\to\infty}\int f(x)\,G_N(\,dx)=\int f(x)\,G_0^T(\,dx) 
\label{(8.27)}
\end{equation}
for all continuous functions~$f$ which vanish outside the cube
$(-T\pi,T\pi)^\nu$.

Indeed, let a continuous function $f$ vanish outside the cube 
$(-T\pi, T\pi)^\nu$ with some $T\ge1$. Put $M=[\frac N{2T}]$. 
Then
\begin{eqnarray*}
\int f(x)G_N(\,dx)&&=\frac{N^\alpha}{L(N)}
\cdot \frac{L(M)}{M^\alpha}
\int f\left(\frac NMx\right)G_M(\,dx)\\
&&=\frac{N^\alpha L(M)}{M^\alpha L(N)}\int f
\left(\frac NMx\right) |K_M(x)|^{-2}\mu_M(\,dx)\\
&&\qquad\qquad \to (2T)^\alpha
\int f(2Tx)|K_0(x)|^{-2}\mu_0(\,dx)\\
&&\qquad\qquad\qquad
=\int f(x)\frac{(2T)^\alpha}{|K_0(\frac x{2T})|^2}\mu_0
\left(\,\frac{dx}{2T}\right)
\quad \textrm{as }N\to\infty,
\end{eqnarray*}
because $f(\frac NMx)|K_M(x)|^{-2}$ vanishes outside the cube
$[-\pi,\pi]^\nu$, the limit relation
$$
f(\frac NMx)|K_M(x)|^{-2}\to f(2Tx)|K_0(x)|^{-2} 
$$ 
holds uniformly, (the function~$K_0(\cdot)^{-2}$ is 
continuous in the cube $[-\pi,\pi]^\nu$), and 
$\mu_M\stackrel{w}{\rightarrow}\mu_0$ as $N\to\infty$. 
Hence relation~(\ref{(8.27)}) holds if we define $G_0^T$ 
as the restriction of the measure
$\frac{(2T)^\alpha}{|K_0(\frac x{2T})|^2}\mu_0
\left(\,\frac{dx}{2T}\right)$ to the cube $(-T\pi,T\pi)^\nu$.
The measures $G_0^T$ appearing in~(\ref{(8.27)}) are 
consistent for different parameters~$T$, i.e.\ $G_0^T$ 
is the restriction of the measure $G_0^{T'}$ to the cube 
$(-T\pi,T\pi)^\nu$ if $T'>T$. This follows from the fact
that $\int f(x)G_0^T(\,dx)=\int f(x)G_0^{T'}(\,dx)$ for
all continuous functions with support in $(-T,T)^\nu$.
We claim that by defining the measure $G_0$ by the relation
$G_0(A)=G_0^T(A)$ for a bounded set $A$ and such number
$T>1$ for which $A\subset (-T\pi,T\pi)^\nu$ we get such a
locally finite measure~$G_0$ for which 
$G_N\stackrel{v}{\rightarrow}G_0$. The above mentioned
vague convergence is a direct consequence of~(\ref{(8.27)})
and the definition of~$G_0$, but to give a complete proof
we have to show that $G_0$ is really a ($\sigma$-additive) 
measure.

Actually it is enough to prove that the restriction of
$G_0$ to the bounded, measurable sets is $\sigma$-additive, 
because it follows then from standard results in measure 
theory that it has a unique $\sigma$-additive extension to 
${\cal B}^\nu$. But this is an almost direct consequence
of the definition of $G_0$. The desired $\sigma$-aditivity 
clearly holds, since if $A=\bigcup\limits_{n=1}^\infty A_n$, 
the set $A$ is bounded, and the sets $A_n$, $n=1,2,\dots$, 
are disjoint, then there is a number $T>1$ such that 
$A\subset (-T\pi,T\pi)^\nu$, the same relation holds for
the sets $A_n$, and the $\sigma$-additivity of $G_0^T$ 
implies that $G_0(A)=\sum\limits_{n=1}^\infty G_0(A_n)$.

As $G_N\stackrel{v}{\rightarrow} G_0$, and
$|K_N(x)|^2\to|K_0(x)|^2$ uniformly in all bounded regions, 
the relation $\mu_N\stackrel{v}{\rightarrow}\bar\mu_0$ 
holds with the measure $\bar\mu_0$ defined as
$\bar\mu_0(A)=\int_A|K_0(x)|^2G_0(\,dx)$, $A\in{\cal B}^\nu$. 
Since
$\mu_N\stackrel{w}{\rightarrow}\mu_0$ the measures~$\mu_0$
and~$\bar\mu_0$ must coincide, i.e.
$$
\mu_0(A)=\int_A |K_0(x)|^2\,G_0(\,dx), \quad A\in{\cal B}^\nu.
$$
Relation~(\ref{(8.4)}) expresses the fact that $\varphi_0$ is 
the Fourier transform of~$\mu_0$.

It remained to prove the homogeneity property~(\ref{(8.3)})
of the measure~$G_0$. For this goal let us extend the 
definition of the measures~$G_N$ given in~(\ref{(8.2)}) to
all non-negative real numbers~$u$. It is easy to see that 
the relation $G_u\stackrel{v}{\rightarrow} G_0$ as 
$u\to\infty$ remains valid. Hence we get for all fixed 
$s>0$ and continuous functions~$f$ with compact support that
\begin{eqnarray*}
\int f(x) G_0(\,dx)&=&\lim_{u\to\infty}\int f(x)\,G_u(\,dx)
=\lim_{u\to\infty}\frac{s^\alpha L(\frac us)}{L(u)}
\int f(sx)G_{\frac us}(\,dx)\\
&=&s^\alpha \int f(sx)G_0(\,dx)=\int f(x)s^\alpha 
G_0 \left(\frac{dx}s\right).
\end{eqnarray*}
This identity implies the homogeneity property~(\ref{(8.3)}) 
of~$G_0$. Lemma~8.1 is proved. \hfill$\qed$

\medskip
The next result is a generalization of Theorem~8.2.

\medskip\noindent
{\bf Theorem~8.2$'$.} {\it Let $X_n$, $n\in\textrm{\BBB Z}_\nu$, 
be a stationary Gaussian field with a correlation function 
$r(n)$ defined in~(\ref{(8.1)}). Let $H(x)$ be a real 
function with the properties $EH(X_n)=0$ and $EH(X_n)^2<\infty$. 
Let us consider the orthogonal expansion
\begin{equation}
H(x)=\sum_{j=1}^\infty c_jH_j(x), \quad \sum c_j^2j!<\infty, 
\label{(8.28)}
\end{equation}
of the function~$H(\cdot)$ by the Hermite polynomials~$H_j$ 
(with leading coefficients~1). Let $k$ be the smallest index 
in this expansion such that $c_k\neq0$. If $0<k\alpha<\nu$ 
for the parameter~$\alpha$ in~(\ref{(8.1)}), and the field 
$Z^N_n$ is defined by the field $\xi_n=H(X_n)$, 
$n\in\textrm{\BBB Z}_\nu$, and formula~(\ref{(1.1)}), then 
the multi-dimensional distributions of the fields $Z_n^N$ 
with $A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$ tend to those of the 
fields $c_kZ^*_n$, $n\in\textrm{\BBB Z}_\nu$, where the 
field $Z^*_n$ is the same as in Theorem~8.2.}

\medskip\noindent
{\it Proof of Theorem~$8.2'$.}\/ Define
$H'(x)=\sum\limits_{j=k+1}^\infty c_jH_j(x)$ and
$Y_n^N=\frac1{A_N}\sum\limits_{l\in B_n^N}H'(X_l)$.
Because of Theorem~8.2 in order to prove Theorem~$8.2'$ 
it is enough to show that
$$
E(Y_n^N)^2\to0 \quad\textrm{as } N\to\infty.
$$
It follows from Corollary~5.5 that 
$EH_j(X_n)H_l(X_m)=\delta_{j,l}j!(EX_nX_m)^j=\delta_{j,l}j!r(n-m)^j$,
where $\delta_{j,l}=0$ if $j\neq l$, and $\delta_{j,l}=1$ 
if $j=l$. Hence
$$
E(Y_n^N)^2=\frac1{A_N^2}\sum_{j=k+1}^\infty  c_j^2j!\,
\sum_{s,t\in B_n^N}[r(s-t)]^j.
$$
Some calculation yields with the help of this identity and 
formula~(\ref{(8.1)}) that
$$
E(Y_n^N)^2=\frac1{A_N^2}\left[O(N^{2\nu-(k+1)\alpha}
L(N)^{k+1})+O(N^\nu)\right]\to0.
$$
(Observe that we imposed the condition $\sum c_j^2j!<\infty$
which is equivalent to the condition $E H(X_n)^2<\infty$.)
Theorem~$8.2'$ is proved. \hfill$\qed$

\medskip
Let us consider a slightly more general version of the 
problem investigated in Theorem~$8.2'$. Take a stationary 
Gaussian random field  $X_n$, $EX_n=0$, $EX_n^2=1$, 
$n\in\textrm{\BBB Z}_\nu$ with a correlation  function 
satisfying relation~(\ref{(8.1)}), and the field 
$\xi_n=H(X_n)$, $n\in\textrm{\BBB Z}_\nu$, subordinated to 
it with a general function $H(x)$ such that $EH(X_n)=0$ and
$EH(X_n)^2<\infty$. We are interested in the large-scale 
limit of such random fields. Take the Hermite 
expansion~(\ref{(8.28)}) of the function~$H(x)$, and let 
$k$ be the smallest such index for which $c_k\neq0$ in the 
expansion~(\ref{(8.28)}). In Theorem~$8.2'$ we solved
this problem if $0<k\alpha<\nu$. We are interested in 
the question what happens in the opposite case when 
$k\alpha>\nu$. Let me remark that in the case 
$k\alpha\ge\nu$ the field $Z^*_n$,
$n\in\textrm{\BBB Z}_\nu$, which appeared in the limit in
Theorem~$8.2'$ does not exist. The Wiener-It\^o integral
defining $Z^*_n$ is meaningless, because the integral 
which should be finite to guarantee the existence of the 
Wiener--It\^o integral is divergent in this case. Next I 
formulate a general result which contains the answer to 
the above question as a special case.

\medskip\noindent
{\bf Theorem~8.5.} {\it Let us consider a stationary Gaussian 
random field $X_n$, $EX_n=0$, $EX_n^2=1$, 
$n\in\textrm{\BBB Z}_n$, with correlation function 
$r(n)=EX_mX_{m+n}$, $m,n\in\textrm{\BBB Z}_\nu$. Take a 
function $H(x)$ on the real line such that $EH(X_n)=0$ and
$EH(X_n)^2<\infty$. Take the Hermite expansion~(\ref{(8.28)}) 
of the function~$H(x)$, and let $k$ be smallest index in 
this expansion such that $c_k\neq 0$. If
\begin{equation}
\sum_{n\in\textrm{\BBB Z}_\nu}|r(n)|^k<\infty, \label{(8.29)}
\end{equation}
then the limit
$$
\lim_{N\to\infty} EZ^N_n(H_l)^2=\lim_{N\to\infty}
N^{-\nu}\sum_{i\in B^N_n}\sum_{j\in B^N_n}r^l(i-j)=\sigma_l^2l!
$$
exists for all indices $l\ge k$, where $Z^N_n(H_l)$ is 
defined in~(\ref{(1.1)}) with $A_N=N^{\nu/2}$, and 
$\xi_n=H_l(X_n)$ with the $l$-th Hermite polynomial $H_l(x)$ 
with leading coefficient~1. Moreover, also the inequality
$$
\sigma^2=\sum_{l=k}^\infty c_l^2l!\sigma_l^2<\infty
$$
holds.

The finite dimensional distributions of the random field 
$Z^N_n(H)$ defined  in~(\ref{(1.1)}) with $A_N=N^{\nu/2}$ 
and $\xi_n=H(X_n)$ tend to the finite  dimensional 
distributions of a random field $\sigma Z^*_n$ with the  
number~$\sigma$ defined in the previous relation, where 
$Z^*_n$, $n\in\textrm{\BBB Z}_\nu$, are independent,
standard normal random variables.}

\medskip
Theorem 8.5 can be applied if the conditions of Theorem~$8.2'$
hold with the only modification that the condition $k\alpha<\nu$
is replaced by the relation $k\alpha>\nu$. In this case the
relation~(\ref{(8.29)}) holds, and the large-scale limit of the 
random field $Z^N_n$, $n\in\textrm{\BBB Z}_\nu$ with normalization
$A_N=N^{\nu/2}$ is a random field consisting of independent
standard normal random variables multiplied with the
number~$\sigma$. There is a slight generalization of
Theorem~8.5 which also covers the case $k\alpha=\nu$. In this
result we assume instead of the condition~(\ref{(8.29)}) that
$\sum\limits_{n\in \bar B_N} r(n)^k=L(N)$ with a slowly varying
function $L(\cdot)$, where
$\bar B_N=\{(n_1,\dots,n_\nu)\in\textrm{\BBB Z}_\nu\colon\;
-N\le n_j\le N,\; 1\le j\le\nu\}$, and some additional
condition is imposed which states that an appropriately defined
finite number  $\sigma^2=\lim\limits_{N\to\infty}\sigma_N^2$, which
plays the role of the variance of the random variables in the
limiting field, exists. There is a similar large scale limit in
this case as in Theorem~8.5, the only difference is that the
norming constant in this case is $A_N=N^{\nu/2}L(N)^{1/2}$. This
result has the consequence that if the conditions of
Theorem~$8.2'$ hold with the only difference that $k\alpha=\nu$
instead of $k\alpha<\nu$,  then the large scale limit exists
with norming constants $A_N=N^{\nu/2}L(N)$ with an  appropriate
slowly varying function~$L(\cdot)$, and it consists of
independent Gaussian random variables with expectation zero.

The proof of Theorem~8.5 and its generalization that we did not
formulate here explicitly appeared in paper~\cite{r3}. I omit 
its proof, I only make some short explanation about it.

In the proof we show that all moments of the random variables
$Z^N_n$ converge to the corresponding moments of the random
variables $Z^*_n$  as $N\to\infty$. The moments of the random
variables $Z_n^N$ can be calculated by means of the diagram
formula if we either rewrite them in the form of a Wiener--It\^o
integral or apply a version of the diagram formula which gives
the moments of Wick polynomials instead of Wiener--It\^o
integrals. In both cases the moments can be expressed explicitly
by means of the correlation function of the underlying Gaussian
random field. The most important step of the proof is to show
that we can select a special subclass of (closed) diagrams,
called regular diagrams in~\cite{r3} which yield the main 
contribution to the moment $E(Z^N_n)^M$, and their contribution 
can be simply calculated. The contribution of all remaining 
diagrams is~$o(1)$, hence it is negligible. For the sake of 
simplicity let us restrict our attention to the case 
$H(x)=H_k(x)$, and let us explain the definition of the 
regular diagrams in this special case. 

If $M$ is an even number, then take the partitions 
$\{k_1,k_2\}$,  $\{k_3,k_4\}$,\dots, $\{k_{M-1},k_M\}$ 
of the set $\{1,\dots,M\}$ to subsets consisting of 
exactly two elements,  to define the regular diagrams. 
They are those (closed)  diagrams for which we can 
choose one of the above partitions in such a way that 
the diagram contains only edges connecting vertices 
from the $k_{2j-1}$-th and $k_{2j}$-th row with  some 
$1\le j\le \frac M2$, where $\{k_{2j-1},k_{2j}\}$ is 
an element of the partition we have chosen. If $M$ is 
an odd number, then there is no regular diagram.

\medskip
In Theorems~8.2 and~$8.2'$ we investigated some very special
subordinated fields. The next result shows that  the same
limiting field as the one in Theorem~8.2 appears in a much
more general situation.

Let us define the field
\begin{equation}
\xi_n=\sum_{j=k}^\infty\frac1{j!}\int e^{i(n,x_1+\cdots+x_j)}
\alpha_j(x_1,\dots,x_j)\,Z_G(\,dx_1)\dots Z_G(\,dx_j),
\quad n\in\textrm{\BBB Z}_\nu, \label{(8.30)}
\end{equation}
where $Z_G$ is the random spectral measure adapted to a Gaussian
field~$X_n$, $n\in\textrm{\BBB Z}_\nu$, with correlation function
satisfying~(\ref{(8.1)}) with $0<\alpha<\frac \nu k$.

\medskip\noindent
{\bf Theorem 8.6.} {\it Let the fields $Z^N_n$ be 
defined by formulae~(\ref{(8.30)}) and~(\ref{(8.1)}) with 
$A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$. The multi-dimensional 
distributions of the fields $Z_n^N$ tend to those of the 
field $\alpha_k(0,\dots,0)Z^*_n$ where the field~$Z^*_n$
is the same as in Theorem~8.2 if the following conditions 
are fulfilled:

\medskip
\begin{description}
\item[(i)]$\alpha_k(x_1,\dots,x_k)$ is a bounded function, 
continuous at the origin, and such that \newline
$\alpha_k(0,\dots,0)\neq0$.
\item[(ii)]
\begin{eqnarray*}
\sum_{j=k=1}^\infty\frac1{j!}\frac{N^{-(j-k)\alpha}}{L(N)^{j-k}}
\int_{ R^{j\nu}}
&&\left|\alpha_j\left(\frac{x_1}N,\dots,\frac{x_j}N\right)\right|^2 
\frac1{N^{2\nu}}
\left|\sum_{j\in B_0^N}e^{i(l/N,x_1+\cdots+x_j)}\right|^2 \\
&&\qquad G_N(\,dx_1)\dots G_N(\,dx_j)\to0,
\end{eqnarray*}
where $G_N$ is defined in~(\ref{(8.2)}).
\end{description}
}

\medskip\noindent
{\it Proof of Theorem 8.6.}\/ The proof is very similar to those of
Theorem~8.2 and~$8.2'$. The same argument as in the proof of
Theorem~$8.2'$ shows that because of condition~(ii) $\xi_n$ can be
substituted in the present proof by the following expression:
$$
\xi_n'=\frac1{k!}\int e^{i(n,x_1+\cdots+x_k)}\alpha_k(x_1,\dots,x_k)
Z_G(\,dx_1)\dots Z_G(\,dx_k), \quad n\in\textrm{\BBB Z}_\nu.
$$
Then a natural modification in the proof of Theorem~8.2 implies
Theorem~8.6. The main point in this modification is that we have to
substitute the measures~$\mu_N$ defined in formula~(\ref{(8.15)}) by the
following measure $\bar\mu_N$:
\begin{eqnarray*}
\bar\mu_N(A)&&=\int_A|K_N(x_1,\dots,x_k)|^2
\left|\alpha_k\left(\frac {x_1}N,\dots,\frac{x_k}N\right)\right|^2
G_N(\,dx_1)\dots G_N(\,dx_k), \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad A\in{\cal B}^{k\nu},
\end{eqnarray*}
and to observe that because of condition~(i) the limit relation
$\mu_N\stackrel{w}{\rightarrow}\mu_0$ implies that
$\bar\mu_N\stackrel{w}{\rightarrow}|\alpha_k(0,\dots,0)|^2\mu_0$.
\hfill$\qed$

\medskip
The main problem in applying Theorem~8.6 is to check conditions~(i)
and~(ii). We remark without proof that any field
$\xi_n=H(X_{s_1+n},\dots,X_{s_p+n})$,
$s_1,\dots,s_p\in\textrm{\BBB Z}_\nu$ and $n\in\textrm{\BBB Z}_\nu$,
for which $E\xi_n^2<\infty$ satisfies condition~(ii). This is 
proved in Remark~6.2 of~\cite{r9}. If the conditions~(i) or~(ii) 
are violated, then a limit of different type may appear. 

Finally we quote such a result without proof. Actually the 
proof is similar to that of Theorem~8.2. At this point the 
general formulation of Lemma~8.3 is  useful. (See~\cite{r24} 
for a proof.) Here we restrict ourselves to the case $\nu=1$. 
The limiting field appearing in this result belongs to the 
class of self-similar fields constructed in Remark~6.5.

Let $a_n$, $n=\dots,-1,0,1,\dots$, be a sequence of real numbers
such that
\begin{equation}
\begin{array}{l}
a_n=C(1)n^{-\beta-1}+o(n^{-\beta-1})\quad \textrm{if } n\ge 0 \\
a_n=C(2)|n|^{-\beta-1}+o(|n|^{-\beta-1})\quad \textrm{if } n< 0
\end{array} 
\qquad -1<\beta<1. \label{(8.31)}
\end{equation}
Let $X_n$, $n=\dots,-1,0,1,\dots$, be a stationary Gaussian
sequence with correlation function
$r(n)=EX_0X_n=|n|^{-\alpha}L(|n|)$, \ $0<\alpha<1$, where
$L(\cdot)$ is a slowly varying function. Define the field
$\xi_n$, $n=\dots,-1,0,1,\dots$, as
\begin{equation}
\xi_n=\sum_{m=-\infty}^\infty a_mH_k(X_{m+n}). \label{(8.32)}
\end{equation}

\medskip\noindent
{\bf Theorem 8.7.} {\it Let a sequence $\xi_n$,
$n=\dots,-1,0,1,\dots$, be defined by~(\ref{(8.31)}) 
and (\ref{(8.32)}). Let
$0<k\alpha<1$, $0<1-\beta-\frac k2\alpha<1$, and let one of
the following conditions be satisfied.

\medskip
\begin{description}
\item[(a)] $0<\beta<1$, and $\sum\limits_{n=-\infty}^\infty a_n=0$.
\item[(b)] $0>\beta>-1$.
\item[(c)] $\beta=0$, $C(1)=-C(2)$, and
$\sum\limits_{n=0}^\infty|a_n+a_{-n}|<\infty$.
\end{description}

\medskip\noindent
Let us define the sequences $Z^N_n$ by formula~(\ref{(1.1)}) 
with $A_N=N^{1-\beta-k\alpha/2}L(N)^{k/2}$ and the above defined
field~$\xi_n$. The multi-dimensional distributions of the
sequences~$Z^N_n$ tend to those of the sequences
$D^{-k}Z^*_n(\alpha,\beta,a,b,c)$, where
\begin{eqnarray*}
Z^*_n(\alpha,\beta,k,b,c)&&=\int\tilde\chi_n(x_1+\cdots+x_k)\\
&&\qquad \left[b|x_1+\cdots+x_k|^\beta+ic|x_1+\cdots+x_k|^\beta
\textrm{\rm sign}\,(x_1+\cdots+x_k)\right] \\
&&\qquad\qquad |x_1|^{(\alpha-1)/2}\cdots |x_k|^{(\alpha-1)/2}\,
W(\,dx_1)\dots W(\,dx_k),
\end{eqnarray*}
$W(\cdot)$ denotes the white noise field, i.e. a random 
spectral measure corresponding to the Lebesgue measure, 
and the constants~$D$, $b$ and~$c$ are defined as 
$D=2\Gamma(\alpha)\cos(\frac\alpha2\pi)$, and

\medskip
\begin{description}
\item[ ]
$b=2[C(1)+C(2)]\Gamma(-\beta)\sin(\frac{\beta+1}2\pi)$, 
and $c=2[C(1)-C(2)]\Gamma(-\beta)\cos(\frac{\beta+1}2\pi)$
in cases~(a) and~(b), and
\item[ ] $b=\sum\limits_{n=-\infty}^\infty a_n$, and $c=C(1)$
in case~(c).
\end{description}
}

\chapter{History of the Problems: Comments}

{\it Chapter 1.}

\medskip\noindent
In statistical physics the problem formulated in this chapter 
appeared at the investigation of some physical models at 
critical temperature. A discussion of this problem and 
further references can be found in the fourth chapter of 
the forthcoming book of Ya.~G.~Sinai~\cite{r33}. (Here and 
in the later part of Chapter~9 we did not change the text 
of the first edition. Thus expressions like forthcoming 
book, recent paper, etc. refer to the time when the first 
version of this Lecture Note appeared.) The first example 
of a limit theorem for partial sums of random variables 
which is considerably  different from the independent 
case was given by M.~Rosenblatt in~\cite{r28}. Further 
results in this direction were proved by R.~L.~Dobrushin, 
H.~Kesten and F.~Spitzer, P.~Major, M.~Rosenblatt and 
M.~S.~Taqqu \cite{r7}, \cite{r8}, \cite{r9}, \cite{r24}, 
\cite{r29}, \cite{r30}, \cite{r34}, \cite{r37}. In most 
of these papers only the one-dimensional case is 
considered, i.e. the case when $R^\nu=R^1$, and it is 
formulated in a different but equivalent way. In most of
these works the joint distribution of the random variables 
$A_N^{-1}\sum\limits_{j=1}^{Nt]}\xi_j$, $0<t<\infty$, was 
considered.

Similar problems also appeared in the theory of infinite 
particle systems. The large-scale limit of the so-called 
voter model and of infinite particle branching Brownian 
motions were investigated in papers \cite{r2}, \cite{r6}, 
\cite{r17}, \cite{r23}. It was proved that in these 
models the limit is a Gaussian self-similar field with a 
non-typical normalization. The investigation of the 
large-scale limit would be very natural for many other 
infinite particle systems, but in most cases this
problem is hopelessly difficult.

The notion of subordinated fields in the present context 
first appeared at Dobrushin~\cite{r7}. It is natural to 
expect that there exists a large class of self-similar 
fields which cannot be obtained as subordinated fields. 
Nevertheless the present techniques are not powerful 
enough for finding them.

The approach  to the problem is different in statistical 
physics. In statistical physics one looks for self-similar 
fields which satisfy some conditions formulated in 
accordance to physical considerations. One tries to 
describe these fields with the help of a power series
which is the Radon--Nykodim derivative of the field with 
respect to a Gaussian field. The deepest result in this 
direction is a recent paper of P.~M.~Bleher and 
M.~D.~Missarov~\cite{r1} where the required formal power 
series is described. This result enables one to calculate 
several critical indices interesting for physicists, but
the task of proving that this formal expression defines 
an existing field seems to be very hard. It is also an 
open problem whether the class of self-similar fields 
constructed via multiple Wiener--It\^o integrals 
contains the non-Gaussian self-similar fields 
interesting for statistical physics. Some experts are 
very skeptical in this respect. The Gaussian 
self-similar fields are investigated in~\cite{r7} 
and~\cite{r32}. A more thorough investigation is 
presented in~\cite{r11}.

The notion of generalized random fields was introduced by 
I.~M.~Gelfand. A detailed discussion can be found in 
the book~\cite{r15}, where the properties of Schwartz 
spaces we need can also be found.

In the definition of generalized fields the class of test
functions~${\cal S}$ can be substituted by other linear 
topological spaces consisting of real valued functions. 
The most frequently considered space, beside the 
space~${\cal S}$, is the space~${\cal D}$ of infinitely 
many times differentiable functions with compact support. 
In paper~\cite{r7} Dobrushin also considered the space
${\cal S}^r\subset{\cal S}$, which consists of the 
functions $\varphi\in{\cal S}$ satisfying the additional 
relation 
$\int {x^{(1)}}^{j_1}\cdots {x^{(\nu)}}^{j_\nu}\varphi(x)\,dx=0$,
provided that $j_1+\cdots+j_\nu<r$. He considered this 
class of test functions, because there are much more 
continuous linear functionals over~${\cal S}^r$ than 
over~${\cal S}$, and this property of~${\cal S}^r$ can 
be exploited in certain investigations. Generally no 
problem arises in the proofs if the space of test
functions~${\cal S}$ is substituted by~${\cal S}^r$ 
or ${\cal D}$ in the definition of generalized fields.

Two generalized fields $X(\varphi)$ and $\bar X(\varphi)$ 
can be identified if 
$X(\varphi)\stackrel{\Delta}{=}\bar X(\varphi)$ for all
$\varphi\in{\cal S}$. Let me remark that this relation 
also implies that the multi-dimensional distributions of 
the random vectors $(X(\varphi_1),\dots,X(\varphi_n))$ 
and $(\bar X(\varphi_1),\dots,\bar X(\varphi_n))$ 
coincide for all $\varphi_1,\dots,\varphi_n\in{\cal S}$. 
As ${\cal S}$ is a linear space, this relation can be 
deduced from property~(a) of generalized fields by 
exploiting that two distribution functions on~$R^n$ 
agree if and only if their characteristic functions 
agree.

Let ${\cal S}'$ denote the space of continuous linear 
functionals over~${\cal S}$, and let 
${\cal A}_{{\cal S}'}$ be the $\sigma$-algebra over 
${\cal S}'$ generated by the sets
$A(\varphi,a)=\{F\colon\;\; F\in{\cal S}';,\,F(\varphi)<a\}$, 
where $\varphi\in{\cal S}$ and $a\in R^1$ are arbitrary. 
Given a probability space  
$({\cal S}',{\cal A}_{{\cal S}'},P)$, a generalized 
field $\bar X=\bar X(\varphi)$ can be defined on it 
by the formula
$\bar X(\varphi)(F)=F(\varphi)$, \ $\varphi\in{\cal S}$, 
and $F\in{\cal S}'$. The following deep result is 
due to Minlos (see e.g.~\cite{r15}).

\medskip\noindent
{\bf Theorem. (Minlos.)} 
\index{Minlos's theorem} 
{\it Let $(X(\varphi),\,\varphi\in{\cal S})$
be a generalized random field. There exists a 
probability measure~$P$ on the measurable space 
$({\cal S},{\cal A}_{{\cal S}'})$ such the 
generalized field
$\bar X=(\bar X(\varphi),\,\varphi\in{\cal S})$ 
defined on the probability space 
$(\bar S,{\cal A}_{{\cal S}'},P)$ by the formula
$\bar X(\varphi)(F)=F(\varphi)$, 
$\varphi\in{\cal S}$, $F\in{\cal S}'$, satisfies the 
relation $X(\varphi)\stackrel{\Delta}{=}\bar X(\varphi)$
for all $\varphi\in{\cal S}$.}

\medskip
The generalized field $\bar X$ has some nice properties. 
Namely property~(a) in the definition of generalized 
fields holds for all $F\in{\cal S}'$. Moreover $\bar X$ 
satisfies the following strengthened version of 
property~(b):

\medskip
\begin{description}
\item[(b$'$)] $\lim\bar X(\varphi_n)=\bar X(\varphi)$ 
in every point $F\in{\cal S}'$ if $\varphi_n\to\varphi$ 
in the topology of~${\cal S}$.
\end{description}

\medskip
Because of this nice behaviour of the field 
$\bar X(\varphi)$ most authors define generalized 
fields as the versions $\bar X$ defined in Minlos' 
theorem. Since we have never needed the extra 
properties of the field~$\bar X$ we have deliberately 
avoided the application of Minlos' theorem in the 
definition of generalized random fields. Minlos' 
theorem heavily depends on some topological properties
of~${\cal S}$, namely that ${\cal S}$ is a so-called 
nuclear space. Minlos' theorem also holds if the space 
of test functions is substituted by ${\cal D}$ or
 ${\cal S}^r$ in the definition of generalized fields.

\medskip
Let us finally remark that Lamperti~\cite{r21} gave 
an interesting characterization of self-similar random 
fields. Let $X(t)$, $t\in R^1$, be a continuous time 
stationary random process, and define the random
process $Y(t)=\frac{X(\log t)}{t^\alpha}$, $t>0$, with 
some $\alpha>0$. Then, as it is  not difficult to see, 
the random processes $Y(t)$, $t>0$, and 
$\frac{Y(ut)}{u^\alpha}$, $t>0$, have the same finite
dimensional distributions for all $u>0$. This can be 
interpreted so that $Y(t)$ is a self-similar process 
with parameter~$\alpha>0$ on the half-line $t>0$. 
Contrariwise, if the finite dimensional distributions 
of the processes $Y(t)$ and $\frac{Y(ut)}{u^\alpha}$,
$t>0$, agree for all $u>0$, then the process
$X(t)=\frac{X(e^t)}{e^{\alpha t}}$, $t\in R^1$, is 
stationary. These relations show some connection 
between stationary and self-similar processes. But 
they have a rather limited importance in the 
investigations of this work, because here we are 
really interested in such random fields which are 
simultaneously stationary and self-similar.

\medskip\noindent
{\it Chapter 2.}

\medskip\noindent
Wick polynomials are widely used in the literature 
of statistical physics. A detailed discussion about 
Wick polynomials can be found in~\cite{r12}. 
Theorems~2A and~2B are well-known, and  they can be 
found in the standard literature. Theorem~2C can be 
found e.g. in Dynkin's book~\cite{r13} (Lemma~1.5). 
Theorem~2.1  is due to Segal~\cite{r31}. It is closely 
related to a result of Cameron and Martin~\cite{r4}. 
The remarks at the end of  the chapter about  the 
content of formula~2.1 are related to~\cite{r25}.

\medskip\noindent
{\it Chapter 3.}

\medskip\noindent
Random spectral measures were independently introduced 
by Cramer and Kolmogorov \cite{r5}, \cite{r20}. They 
could have been  introduced by means of Stone's theorem 
about the spectral  representation of one-parameter 
groups of unitary operators.  Bochner's theorem can be 
found in any standard book on  functional analysis, 
the proof of the Bochner--Schwartz  theorem can be 
found in~\cite{r15}. Let me remark that the same result 
holds true if the space of test functions~${\cal S}$ is 
substituted by~${\cal D}$.

There is an object, called the fractional Brownian motion, 
which is a popular topic of many investigations, and which
can be studied by means of the method of this chapter.
In particular, the results of Chapter~3 imply their
existence. A fractional Brownian motion with Hurst 
parameter $H$, $0<H<1$, is a Gaussian process $X(t)$, 
$t\ge0$, with continuous trajectories and zero expectation, 
i.e. $EX(t)=0$ for all $t\ge0$, and with covariance 
function 
$R_H(s,t)=EX(s)X(t)=\frac12(s^{2H}+t^{2H}-|t-s|^{2H})$ 
for all $0\le s,t<\infty$. 
\index{fractional Brownian motion} \index{Hurst parameter}
Let us explain that the  
correlation of a fractional Brownian motion  has a natural 
representation as the correlation function of the 
discretized version of an appropriately defined Gaussian 
stationary generalized self-similar field. In the subsequent
argument the representation of (generalized) stationary
Gaussian fields turned out to be very useful.

To find this representation observe that a fractional Brownian 
motion with Hurst parameter~$H$ has the self-similarity property 
$EX(as)X(at)=a^{2H}EX(s)X(t)$ for all $a>0$, and simple
calculation shows that it also has the following stationary
increments property: $E[X(s+u)-X(u)][X(t+u)-X(u)]=EX(s)X(t)$
for all $0\le s,t,u<\infty$. Hence we can construct a fractional
Brownian motion $X(t)$ by defining first an appropriate stationary, 
Gaussian generalized self-similar field $\bar X(\varphi)$, 
$\varphi\in{\cal S}_1$ in the space of the real valued 
functions of the Schwartz space, and then by extending it to 
a larger parameter set (of functions), containing the indicator 
functions $\chi_{[0,t]}$ of the intervals $[0,t]$ for all $t\ge0$.
Finally we define the process $X(t)$ as $X(t)=\bar X(\chi_{[0,t]})$.

More explicitly, let us define for a parameter $\alpha$ the
stationary generalized Gaussian field $\bar X(\varphi)$, 
$\varphi\in{\cal S}^1$, with zero expectation and spectral 
density $|u|^{-2\alpha}$, i.e. put 
$E\bar X(\varphi)\bar X(\psi)=\int 
\tilde\varphi(u)\bar{\tilde\varphi}(u)|u|^{-2\alpha}\,du$,
and introduce its (discretized) extension to a function 
space containing the functions $\chi_{[0,t]}$ for all $t>0$. 
Then we have
$$
E\bar X(\chi_{[0,s]})\bar X(\chi_{[0,t]})=
\int \tilde\chi_{[0,s]}(u)\bar{\tilde \chi}_{[0,t]}(u)
|u|^{-2\alpha}\,du
=\int \frac{e^{isu}-1}{iu}\frac{e^{-itu}-1}{-iu}
|u|^{-2\alpha}\,du,
$$
provided that these integrals are convergent.  

The above defined generalized fields exist if $2\alpha>-1$, 
and their discretized extension exists if $-1<2\alpha<1$. The 
first condition is needed to guarantee that the singularity 
of the integrand in the formula expressing the covariance 
function is not too strong in the origin, and the second 
condition is needed to guarantee that the singularity of 
this integrand is not too strong at the infinity even in 
the discretized case.

Simple calculation shows that the covariance function of 
the above defined random field satisfies the identity 
$E\bar X(\varphi_a)\bar X(\psi_a)=a^{-(1+2\alpha)}
E\bar X(\varphi)\bar X(\psi)$, with the functions
$\varphi_a(x)=\varphi(ax)$, $\psi_a(x)=\psi(ax)$, and 
similarly, we have $EX(as)X(at)=a^{(1+2\alpha)}EX(s)X(t)$ 
for  all $a>0$. Besides, the Gaussian stochastic process 
$X(t)$, $t>0$, has stationary increments, i.e. 
$E[X(s+u)-X(u)][X(t+u)-X(u)]=EX(s)X(t)$
for all $0\le s,t,u<\infty$. This follows from its 
construction with the help of a stationary Gaussian 
random field. 

The above calculations imply that with the choice 
$\alpha=H-1/2$ we get the covariance function of a 
fractional Brownian motion with Hurst parameter $H$ 
for all $0<H<1$, more precisely the correlation 
function of this process multiplied by an appropriate 
constant. Indeed, it follows from the stationary 
increments property of the process that 
$E(X(t)-X(s))^2=EX(t-s)^2$, if $t\ge s$, and the 
self-similarity property of this process implies that 
$EX(s)X(t)=\frac12[EX(s)^2+EX(t)^2-E(X(t)-X(s))^2]
=\frac12 EX(1)^2[s^{2H}+t^{2H}-|t-s|^{2H}]$.

Actually the results of Chapter~3 also provide a 
representation of this process by means of an integral 
with respect to a random spectral measure. This 
representation has the form
$$
X(t)=\int\frac {e^{itu}-1}{iu}|u|^{-H+1/2}Z(\,du), \quad t>0,
$$
with the random spectral measure $Z(\cdot)$ corresponding to
the Lebesgue measure on the real line. Here we omit the proof 
that such a stochastic process also has a version with 
continuous trajectories.

\medskip\noindent
{\it Chapter 4.}

\medskip\noindent
The stochastic integral defined in this chapter is a 
version of  that introduced by~It\^o in~\cite{r18}. 
This modified integral  first appeared in Totoki's 
lecture note~\cite{r38} in a special  form. Its 
definition is a little bit more difficult than the 
definition of the original stochastic integral 
introduced by  It\^o, but it has the advantage that 
the effect of the shift  transformation can be better 
studied with its help. Most  results of this chapter 
can be found in Dobrushin's paper~\cite{r7}. The 
definition of Wiener--It\^o integrals in the case 
when the spectral measure may have atoms is new. In 
the new version of this lecture note I worked out 
many  arguments in a more detailed form than in the 
old text. In  particular, in Lemma~4.1 I gave a much 
more detailed explanation of  the statement that all 
kernel functions of Wiener--It\^o integrals can be 
well approximated by simple functions.

\medskip\noindent
{\it Chapter 5.}

\medskip\noindent
Proposition~5.1 was proved for the original Wiener--It\^o 
integrals by It\^o in~\cite{r18}. Lemma~5.2 contains a 
well-known formula about Hermite  polynomials. The main 
result of this chapter, Theorem~5.3, appeared in 
Dobrushin's work~\cite{r7}. The proof given there is 
not complete. Several non-trivial details are omitted. 
I felt even necessary to present a more detailed proof 
in this note when I wrote down its new version. 
Theorem~5.3 is closely related to Feynman's diagram 
formula. The result of Corollary~5.5 was already known 
at the beginning of the XX. century. It was proved 
with the help of some formal manipulations. This formal 
calculation was justified by Taqqu in~\cite{r35} with 
the help of some deep inequalities. In the new version 
of this note I formulated a more general result than 
in the older one. Here I gave a formula about the 
expectation of products of Wick polynomials and not only 
of Hermite polynomials.

I could not find results similar to Corollaries~5.6 
and~5.7 in the literature of probability theory. On the 
other hand, such results are well-known in statistical 
physics, and they play an important role in constructive 
field theory. A sharpened form of these results is
Nelson's deep hypercontractive inequality~\cite{r27}, 
which I  formulate below.
\index{Nelson's hypercontractive inequality}

Let $X_t$, $t\in T$, and $Y_{t'}$, $t'\in T'$ be two sets 
of jointly Gaussian random variables on some probability 
spaces $(\Omega,{\cal A},P)$ and $(\Omega,{\cal A}',P')$. 
Let ${\cal H}_1$ and ${\cal H}_1'$ be the Hilbert 
spaces generated by the finite linear combinations 
$\sum c_jX_{t_j}$ and $\sum c_jY_{t'_j}$. Let us define 
the $\sigma$-algebras ${\cal B}=\sigma(X_t,\,t\in T)$ 
and ${\cal B}'=\sigma(Y_{t'},\,t'\in T')$ and the 
Banach spaces $L_p(X)=L_p(\Omega,{\cal B},P)$,
$L_p(Y)=L_p(\Omega',{\cal B}',P')$, $1\le p\le\infty$. 
Let $A$ be a linear transformation from ${\cal H}_1$ to 
${\cal H}_1'$ with norm not exceeding~1. We define an 
operator $\Gamma(A)\colon L_p(X)\to L_{p'}(Y)$ for all 
$1\le p,p'\le\infty$ in the following way. If $\eta$ 
is a homogeneous polynomial of the variables~$X_t$,
$$
\eta=\sum C_{j_1,\dots,j_s}^{t_1,\dots,t_s} 
X^{j_1}_{t_1}\cdots X^{j_s}_{t_s}, \quad t_1,\dots,t_s\in T,
$$
then
$$
\Gamma(A)\colon\!\eta\!\colon=
\sum C_{j_1,\dots,j_s}^{t_1,\dots,t_s}
\colon\!(AX_{t_1})^{j_1}\cdots (AX_{t_s})^{j_s}\!\colon.
$$
It can be proved that this definition is meaningful, i.e.
$\Gamma(A)\colon\!\eta\!\colon$ does not depend on the 
representation of $\eta$, and $\Gamma(A)$ can be extended 
to a bounded operator from $L_1(X)$ to $L_1(Y)$ in a unique 
way. This means in particular that $\Gamma(A)\xi$ is defined 
for all $\xi\in L_p(X)$, $p\ge1$. Nelson's hypercontractive
inequality says the following. Let $A$ be a contraction 
from ${\cal H}_1$ to ${\cal H}_1'$. Then $\Gamma(A)$ is a 
contraction from $L_q(X)$ to $L_p(Y)$ for $1\le q\le p$ 
provided that
\begin{equation}
\|A\|\le\left( \frac{q-1}{p-1}\right)^{1/2}. \label{($+$)}
\end{equation}
If~(\ref{($+$)}) does not hold, then $\Gamma(A)$ is not a 
bounded operator from $L_q(X)$ to $L_p(Y)$.

A further generalization of this result can be found 
in~\cite{r16}.

The following discussion may help to understand the 
relation between Nelson's hypercontractive inequality 
and Corollary~5.6. Let us apply Nelson's inequality in 
the special case when $(X_t,\,t\in T)=(Y_{t'},\,t'\in T')$ 
is a stationary Gaussian field with spectral measure~$G$, 
$q=2$, $p=2m$ with some positive integer~$m$,
$A=c\cdot\textrm{Id}$, where $\textrm{Id}$ denotes 
the identity operator, and $c=(2m-1)^{-1/2}$. Let 
${\cal H}^c$ and ${\cal H}^c_n$ be the complexification 
of the real Hilbert spaces ${\cal H}$ and ${\cal H}_n$
defined in Chapter~2. Then 
$L_2(X)={\cal H}^c={\cal H}^c_0+{\cal H}_1^c+\cdots$ 
by Theorem~2.1 and formula~2.1. The operator 
$\Gamma(c\cdot\textrm{Id}\,)$ equals 
$c^n\cdot\textrm{Id}$ on the subspace ${\cal H}_n^2$. 
If $h_n\in{\cal H}^n_G$, then $I_G(h_n)\in {\cal H}_n$, 
hence the application of Nelson's inequality for the 
operator $A=c\cdot\textrm{Id}$ shows that
$$
\left(EI_G(h_n)^{2m}\right)^{1/2m}
=c^{-n}\left(E(\Gamma(c\cdot\textrm{Id})
I_G(h_n))^{2m}\right)^{1/2m}
\le c^{-n}\left(EI_G(h_n)^2\right)^{1/2}
$$
i.e.
$$
EI_G(h_n)^{2m}\le c^{-2nm}\left(EI_G(h_n)^2\right)^m
=(2m-1)^{mn}\left(EI_G(h_n)^2\right)^m.
$$
This inequality is very similar to the second inequality 
in Corollary~5.6, only the multiplying constants are 
different. Moreover, for large~$m$ these multiplying 
constants are near to each other. I remark that the 
following weakened form of Nelson's inequality could
be deduced relatively easily from Corollary~5.6. Let
$A\colon\;{\cal H}_1\to{\cal H}'_1$ be a contraction 
$\|A\|=c<1$. Then there exists a $\bar p=\bar p(c)>2$ 
such that $\Gamma(A)$ is a bounded operator from 
$L_2(X)$ to $L_p(Y)$ for $p<\bar p$. This weakened 
form of Nelson's inequality is sufficient in many
applications.

\medskip\noindent
{\it Chapter 6.}

\medskip\noindent
Theorems~6.1,~6.2 and Corollary~6.4 were proved by 
Dobrushin in~\cite{r7}. Taqqu proved similar results 
in~\cite{r36}, but he gave a different representation. 
Theorem~6.6 was proved by H.~P.~Mc.Kean in~\cite{r26}.
The proof of the lower bound uses some ideas 
from~\cite{r14}. Remark~6.5 is from~\cite{r24}. As 
Proposition~6.3 also indicates, some non-trivial 
problems about the convergence of certain integrals 
must be solved when constructing self-similar fields. 
Such convergence problems are common in statistical 
physics. To tackle such problems the so-called power 
counting method (see e.g.~\cite{r22}) was worked out.
This method could also be applied in this chapter. 
Part~(b) of Proposition~6.3 implies that the 
self-similarity parameter~$\alpha$ cannot be chosen 
in a larger domain in Corollary~6.4. One can ask
about the behaviour of the random variables $\xi_j$ 
and $\xi(\varphi)$ defined in Corollary~6.4 if the 
self-similarity parameter~$\alpha$ tends to the 
critical value~$\frac\nu2$. The variance of the 
random variables~$\xi_j$ and $\xi(\varphi)$ tends
to infinity in this case, and the fields $\xi_j$,
$j\in\textrm{\BBB Z}_\nu$, and $\xi(\varphi)$, 
$\varphi\in{\cal S}$, tend, after an appropriate 
renormalization, to a field of independent normal 
random variables in the discrete, and to a white
noise in the continuous case. The proof of these 
results with a more detailed discussion  
appeared in~\cite{r10}.

In a recent paper~\cite{r19} Kesten and Spitzer have 
proved a limit theorem, where the limit field is a 
self-similar field which seems not to belong to 
the class of self-similar fields constructed in 
Chapter~6. (We cannot however, exclude the 
possibility that there exists some self-similar 
field in the class defined in Theorem~6.2 with 
the same distribution as this field, although it 
is given by a completely different form.) This 
self-similar field constructed by Kesten and
Spitzer is the only rigorously constructed 
self-similar field known for us that does not 
belong to the fields constructed in Theorem~6.2.
I describe this field, and then I make some 
comments.

Let $B_1(t)$ and $B_2(t)$, $-\infty<t<\infty$, be two 
independent Wiener processes. (We say that $B(t)$ is 
a Wiener process on the real line if $B(t)$, $t\ge0$, 
and $B(-t)$, $t\ge0$, are two independent Wiener 
processes.) Let $K(x,t_1,t_2)$, $x\in R^1$, $t_1<t_2$, 
denote the local time of the process $B_1$ at the 
point $x$ in the interval $[t_1,t_2]$. The 
one-dimensional field
$$
Z_n=\int K(x,n,n+1)\,B_2(\,dx), 
\quad n=\dots,-1,0,1,\dots,
$$
where the integral in the last formula is an It\^o 
integral, is a stationary self-similar field with 
self-similarity parameter~$\frac34$.

To see the self-similarity property one has to observe 
that
$$
K(\lambda^{1/2}x,\lambda t_1,\lambda t_2)
\stackrel{\Delta}{=}\lambda^{1/2}
K(x,t_1,t_2) \quad\textrm{for all }x
\in R^1,\;\; t_1<t_2, \textrm{ and }
\lambda>0
$$
because of the relation
$B_1(\lambda u)\stackrel{\Delta}{=}\lambda^{1/2}B_1(u)$. 
Hence
$$
\sum_{j=0}^{n-1}Z_j\stackrel{\Delta}{=}n^{1/2}
\int K(n^{-1/2}x,0,1)B_2(\,dx)\stackrel{\Delta}{=} 
n^{3/4}\int K(x,0,1)\,B_2(\,dx)=n^{3/4}Z_0.
$$
The invariance of the multi-dimensional distributions 
of the  field~$Z_n$ under the 
transformation~(\ref{(1.1)}) can be seen similarly.

To see the stationarity of the field~$Z_n$ we need 
the following two observations.

\medskip
\begin{description}
\item[(a)]$ K(x,s,t)\stackrel{\Delta}{=}K(x+\eta(s),0,t-s)$ 
with $\eta(s)=-B_1(-s)$. (The form of $\eta$ is not 
important for us. What we need is that the pair $(\eta,K)$ 
is independent of $B_2$.)
\item[(b)] If $\alpha(x)$, $-\infty<x<\infty$, is a 
process independent of $B_2$, then
$$
\int \alpha(x+u)B_2(\,dx)\stackrel{\Delta}{=}
\int\alpha(x) B_2(\,dx) \quad\textrm{for all }u\in R^1.
$$
\end{description}

\medskip
It is enough to show, because of Property~(a) that
$$
\int K(x+\eta(s),0,t-s)\,B_2(\,dx)\stackrel{\Delta}{=}
\int K(x,0,t-s)\,B_2(\,dx).
$$
This relation follows from property~(b), because the 
conditional distributions of the left and right-hand 
sides agree under the condition $\eta(s)=u$, $u\in R^1$.

The generalized field version of the above field~$Z_n$ 
is the field
$$
Z(\varphi)=-\int \left[K(x,0,t)
\frac{d\varphi}{dt}\,dt\right] B_2(\,dx),
\quad \varphi\in{\cal S}.
$$
To explain the analogy between the field $Z_n$ and 
$Z(\varphi)$ we remark that the kernel of the integral 
defining~$Z_n$ can be written, at least formally, as
$$
K(x,n,n+1)=\int\chi_{[n,n+1)}(u)\frac d{du}K(x,n,u)\,du,
$$
although $K$ is a non-differentiable function. 
Substituting the function  $\chi_{[n,n+1)}$ by 
$\varphi\in{\cal S}$, and integrating by parts (or 
precisely, considering $\frac d{du}K$ as  the 
derivative of a distribution) we get the above 
definition of $Z(\varphi)$.

Using the same idea as before, a more general class of 
self-similar fields can be constructed. The integrand 
$K(x,n,n+1)$ can be substituted by the local time of any 
self-similar field with stationary increments which is 
independent of $B_2$. Naturally, it must be clarified 
first that this local time really exists. One could 
enlarge this class also by integrating with respect to 
a self-similar field with stationary increments, 
independent of~$B_1$. The integral with respect to a 
field independent of the field $K(x,s,t)$ can be 
defined without any difficulty.

\medskip
There seems to be no natural way to represent the above 
random fields as random fields subordinated to a Gaussian 
random field. On the other hand, the local times 
$K(x,s,t)$ are measurable with respect to~$B_1$, they 
have finite second moments, therefore they can be 
expressed by means of multiple Wiener--It\^o integrals 
with respect to a white noise field. Then the 
process $Z_n$ itself can also be represented via 
multiple Wiener--It\^o integrals. It would be 
interesting to know whether the above defined 
self-similar fields, and probably a larger class of
self-similar fields, can be constructed in a simple 
natural way via multiple Wiener--It\^o integrals with 
the help of a randomization.

\medskip\noindent
{\it Chapter 7.}

\medskip\noindent
The definition of Wiener--It\^o integrals together with 
the proof of Theorem~7.1 and Proposition~7.3 were given 
by It\^o in~\cite{r18}. Theorem~7.2 is proved in Taqqu's 
paper~\cite{r37}. He needed this result to show that the 
self-similar fields defined in~\cite{r9} by means of
Wiener--It\^o integrals coincide with the self-similar 
fields defined in~\cite{r37} by means of modified
 Wiener--It\^o integrals.

\vfill\eject

\noindent
{\it Chapter 8.}

\medskip\noindent
The results of this chapter, with the exception of 
Theorems~8.5 and~8.7 are proved in~\cite{r9}. 
Theorem~8.5 is proved in~\cite{r3} and Theorem~8.7 
in~\cite{r24}. The latter paper was strongly 
motivated by~\cite{r29}. Lemma~8.3 is formulated 
in a more general form than Lemma~3 in~\cite{r9}. 
The  present formulation is more complicated, but 
it is more useful in some applications. Let me 
explain this in more detail. The difference 
between the original and the present formulation 
of this lemma is that here we allow that the 
integrand~$K_0$ in the limiting stochastic integral 
is discontinuous on a small subset of~$R^{k\nu}$, 
and the functions~$K_N$ may not converge on this set. 
This freedom can be exploited in some applications. 
Indeed, let us consider e.g.\ the self-similar 
fields constructed in Remark~6.5. In case $p<0$ the 
integrand in the formula expressing these fields is 
not continuous on the hyperplane $x_1+\cdots+x_n=0$. 
Hence, if we want to prove limit theorems where 
these fields appear as the limit, and this happens 
e.g. in Theorem~8.7 then we can apply Lemma~8.3, 
but not its original version, Lemma~3 in~\cite{r9}.

The example for non-central limit theorems given by 
Rosenblatt in~\cite{r28} and its generalization by 
Taqqu in~\cite{r34} are special cases of Theorem~8.2. 
In these papers only the special case $H_2(x)=x^2-1$ 
is considered. Later Taqqu~\cite{r37} proved a result 
similar to Theorem~$8.2'$, but he needed more 
restrictive conditions. The observation that 
Theorem~$8.2'$ can be deduced from Theorem~8.2 is 
from Taqqu~\cite{r34}.

The method of~\cite{r28} and~\cite{r34} does not apply for 
the proof of Theorem~8.2 in the case of $H_k(x)$, $k\ge3$.  
In these papers it is proved that the moments of the random 
variables $Z_n^N$ converge to the corresponding moments of 
$Z^*_n$. (Actually a different but equivalent statement is 
established in these papers.) This convergence of the
moments implies the convergence
$Z_n^N\stackrel{{\cal D}}{\rightarrow} Z^*_n$ if and 
only if the distribution of $Z^*_n$ is uniquely determined 
by its moments.

Theorem~6.6 implies that the $2n$-th moment of a $k$-fold
Wiener--It\^o integral behaves similarly to the $2kn$-the 
moment of a Gaussian random variable with zero expectation, 
it equals  $e^{(kn\log n)/2+O(n)}$. Hence some results about
the so-called moment problem show that the distribution of 
a $k$-fold Wiener--It\^o integral is determined by its 
moments only for $k=1$ and $k=2$. Therefore the method of 
moments does not work in the proof of Theorem~8.2 for 
$H_k(x)$, $k\ge3$.

Throughout Chapter~8 I have assumed that the correlation 
function of the underlying Gaussian field to which our 
fields  are subordinated satisfies formula~(\ref{(8.1)}). 
This assumption  seems natural, since it implies that the 
spectral measure of  the Gaussian field satisfies Lemma~8.1, 
and such a condition is  needed when $Z_{G_N}$ is 
substituted by~$Z_{G_0}$ in the limit.  It can be asked 
whether in Theorem~8.2 formula~8.1 can be substituted by 
the weaker assumption that the spectral measure of the 
Gaussian field satisfies Lemma~8.1. This question was  
investigated in Section~4 of~\cite{r9}. The investigation 
of the moments shows that the answer is negative. The 
reason for it is that the validity of Lemma~8.1, unlike 
that of Theorem~8.2, does not depend on whether the 
spectral measure~$G$ has large singularities outside 
the origin or not. The discussion in~\cite{r9} also 
shows that the Gaussian case, that is the case when 
$H_k(x)=H_1(x)=x$ in Theorem~8.2, is considerably 
different from the non-Gaussian case. A forthcoming 
paper of M.~Rosenblatt~\cite{r30} gives a better 
insight into the above question.

The limiting fields appearing in Theorem~8.2 and~8.6 
belong to a special subclass of the self-similar fields 
defined in Theorem~6.2. These results  indicate that the
 self-similar fields defined in formula~(\ref{(6.5)}) 
have a much greater range of attraction if the homogeneous 
function~$f_n$ in~(\ref{(6.5)}) is the constant function. 
The reason for the particular behaviour of these fields is 
that the constant function is analytic, while a general 
homogeneous function typically has a singularity at the 
origin. A more detailed discussion about this problem 
can be found in~\cite{r24}.

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\end{thebibliography}

\backmatter

\printindex




\extrachap{Acronyms}

\begin{description}

\item
[${\cal D}$] The space of infinitely differentiable functions 
with compact support
\item            
[${\cal D}'$] The space of generalized functions on the space 
of test function ${\cal D}$
\item
[$d(A)$] The diameter of the set $A$
\item
[$\textrm{Exp}\,{\cal H}_G$ and $\textrm{Exp}\,{\cal K}_\mu$] 
The Fock space
\item
[$G(\cdot)$] The spectral measure of a stationary discrete or 
generalized random field 
\item
[$G_n\stackrel{v}{\rightarrow}G_0$] The vague convergence of
the locally finite measures $G_n$ to the locally finite
measure $G_0$
\item
[$H_n(x)$] The Hermite polynomial of order $n$ with leading 
coefficient 1
\item
[${\cal H}$] The (real) Hilbert space of square-integrable random
variables measurable with respect to the $\sigma$-algebra generated
by the random variables of a previously defined Gaussian random field
\item
[${\cal H}_1$] The smallest subspace of the Hilbert space ${\cal H}$
containing the elements of the underlying Gaussian field 
\item
[${\cal H}_{\le n}$]  
The smallest subspace of the Hilbert space ${\cal H}$ containing 
the polynomials of order less than or equal to $n$ of the
random variables in the underlying Gaussian field
\item
[${\cal H}_n$] The orthogonal completion of the subspace
${\cal H}_{\le n-1}$ in the Hilbert space ${\cal H}_{\le n}$
\item
[$h_\gamma(\cdot)$] The kernel function of the Wiener--It\^o 
integral appearing in the diagram formula and depending 
on the diagram~$\gamma$
\item
[${\bar{\cal H}}^n_G$] The space of functions which can be the
kernel function of an $n$-fold Wiener--It\^o integral with
respect to a random spectral measure $Z_G$ with 
spectral measure~$G$
\item
[${\cal H}^n_G$] The subspace of ${\bar{\cal H}}^n_G$
consisting of symmetric functions 
\item
[${\hat{\bar{\cal H}}}^n_G$] The subspace of 
${\bar{\cal H}}^n_G$ consisting of simple functions 
\item
[${\hat{\cal H}}^n_G$] The subspace of ${\cal H}^n_G$ 
consisting of simple (and symmetric) functions 
\item
[$I_G(f_n)$] The normalized Wiener--It\^o integral of the
kernel function~$f_n$ of $n$ variables with respect to
the random spectral measure $Z_G$
\item
[$\bar{{\cal K}}_\mu^n$] The class of function which can be
the kernel function of an $n$-fold Wiener--It\^o integral with
respect to a random orthogonal measure $Z_\mu$
\item
[${\cal K}_\mu^n$] The subspace of ${\bar{\cal K}}_\mu^n$ 
consisting of symmetric functions
\item
[$\hat{\bar{{\cal K}}}_\mu^n$] The set of simple 
functions appearing in the definition of $n$-fold 
Wiener--It\^o integrals with respect to a random orthogonal
measure $Z_\mu$
\item
[${\cal K}$] The Hilbert space of square integrable random
variables measurable with respect to the $\sigma$-algebra
generated by the random variables $Z_\mu(A)$ of a random
orthogonal measure $Z_\mu$
\item
 [${\cal K}_{\le n}$]
The subspace of ${\cal K}$ generated by the
polynomials of the random variables $Z_\mu(A)$ of the 
orthogonal random field $Z_\mu$ which have order less than 
or equal to $n$
\item
[${\cal K}_n$] The orthogonal completion of the subspace
${\cal K}_{\le n-1}$ in the Hilbert space ${\cal K}_{\le n}$
\item
[$\colon\!P(\xi_1,\dots,\xi_n)\!\colon$] The Wick polynomial
corresponding to the polynomial \hfill\break 
$P(x_1,\dots,x_n)$ and Gaussian
random vector $(\xi_1,\dots,\xi_n)$
\item 
[${\cal S}$] The class of test functions in the
Schwartz space
\item
[${\cal S}^c$] The class of complex number valued test functions 
in the Schwartz space
\item
[${\cal S}'$]  
The Schwartz space of generalized functions
\item
[$S_{\nu-1}$] The $\nu$-dimensional unit sphere
\item
[$\textrm{Sym}\,f$] The symmetrization of the function $f$
\item
[$T_m$ and $T_t$] The shift operator with parameter 
$m\in\textrm{\BBB Z}_\nu$ and $t\in R^\nu$
\item
[$X(\varphi)$] The value of the generalized field $X(\cdot)$
at the test function $\varphi$ 
\item
[$Z_G(\cdot)$] The (Gaussian) random spectral measure corresponding
to the spectral measure $G$
\item
[$Z(\,dx)$] The (Gaussian) random spectral measure whose spectral 
measure is $\frac1{2\pi}$ times the Lebesgue measure on $[-\pi,\pi)$
\item
[$Z_\mu(\cdot)$] The random orthogonal measure corresponding to
the measure $\mu$
\item
[$\textrm{\BBB Z}_\nu$] The set of lattice points in the 
$\nu$-dimensional space with integer coordinates

\item
[$\Gamma(n_1,\dots,n_m)$] The space of diagrams in the diagram formula
\item
[$\bar\Gamma$] The space of closed diagrams
\item
[$|\gamma|$] The number of edges in a diagram $\gamma$
\item 
[$\mu_n\stackrel{w}{\rightarrow}\mu$] The weak convergence of
the probability measures $\mu_n$ to the probability measure $\mu$
\item
[$\xi_N\stackrel{{\cal D}}{\rightarrow}\xi_0$] The convergence
of the random variables $\xi_N$ to the random variable $\xi_0$
in distribution, i.e. the weak convergence of the distributions 
of $\xi_N$ to the distribution of $\xi_0$
\item
[$\Pi_n$] The group of permutations of the set $\{1,\dots,n\}$
\item
[$\rho_p(\mu,\nu)$] The Prokhorov metric of the probability
measures $\mu$ and $\nu$
\item
[$\chi_A(\cdot)$] The indicator function of the set $A$.
\item
[$\tilde\chi_n(x)$] The Fourier transform of the indicator function
of the unit cube \hfill\break 
$\prod\limits_{p=1}^\nu [n^{(p)},n^{(p)}+1)$, where 
$n=(n^{(1)},\dots,n^{(p)})$
\item
[$\ominus$] The orthogonal completion of a subspace
of a Hilbert space
\item
[$\tilde{}$] Fourier transform
\item
[$*$] Convolutions 
\item
[$\stackrel{\Delta}{=}$] Identity in distribution
\item
[$\Rightarrow$] Stochastic convergence
\item
[$\int'$] Wiener--It\^o integral with respect to a 
random orthogonal measure
\item
[$\lbrack x\rbrack$] Integer part of a real number $x$

\end{description}

\vfill\eject

{\it Abstract:}

Chapter 1

\medskip\noindent
We formulate the main problems discussed in this paper together with
the most important notions needed in their discussion. In particular,
we introduce the notion of generalized random fields and also explain
at a heuristic level why their introduction is useful for us. We 
finish this chapter with a sub-chapter that contains a short summary 
about some useful results in the theory of generalized functions.

\medskip

Chapter 2

\medskip\noindent
We introduce the notion of Wick polynomials which are natural 
multi-variate versions of Hermite polynomials. We present their
most important properties, and with their help we give a 
decomposition of the Hilbert space of square integrable random
variables measurable with respect to a stationary Gaussian
random field to the direct sum of orthogonal, shift invariant 
subspaces.

\medskip

Chapter 3

\medskip\noindent
We present the correlation function of a Gaussian stationary field
as the Fourier transform of a spectral measure and construct
with its help a (Gaussian) random spectral measure. Then we express 
a stationary Gaussian field itself as the Fourier transform of 
this random spectral measure. We also describe the most important
properties of spectral and random spectral measures. The proofs
heavily depend on a classical result of analysis about the
representation of so-called positive definite functions as the
Fourier transform of positive measures and on its version about 
generalized functions. Hence we finish this chapter with a 
sub-chapter where we discuss these results, called Bochner and
Bochner--Schwartz theorems in the literature.

\medskip

Chapter 4

\medskip\noindent
Here we introduce the multiple Wiener--it\^o integrals with respect
to a Gaussian random spectral measure and prove some important
results about them.

\medskip

Chapter 5

\medskip\noindent
Here we prove the most important result about multiple Wiener--It\^o
integrals, the so-called diagram formula together with some of its
consequences. In the diagram formula we rewrite the product of
Wiener--It\^o integrals in the form of a sum of Wiener--It\^o 
integrals and also give a formula (with the help of some diagrams) 
about the calculation the kernel-functions of the integrals 
appearing in this sum.

\medskip

Chapter 6

\medskip\noindent
We give a complete characterization of the so-called subordinated
random fields of a stationary Gaussian random field. This result
enables us to construct new, non-trivial (subordinated) 
self-similar random fields, i.e. such random fields which may 
appear as the limit random field in limit theorems. To tell 
whether the formulas defining these subordinated random fields 
are meaningful or not we have to decide whether certain classical 
integrals are convergent or divergent. Hence this chapter 
contains some results in this direction.

\medskip

Chapter 7.

\medskip\noindent
Here we discuss the original Wiener--It\^o integrals with respect
to a random orthogonal measure. We give their most important
properties and also present some results about their relation to
the Wiener--It\^o integrals with respect to a random spectral
measure and to the classical It\^o integrals of stochastic
processes.

\medskip

Chapter 8.

\medskip\noindent
Here we present some non-trivial limit theorems where the
limit is a non-Gaussian self-similar field. The results of
the previous chapters may explain at a heuristic level 
why such results should hold. But a rigorous proof demands
much extra work whose consequences may be interesting in 
themselves.

\medskip

Chapter 9.

\medskip\noindent
Here we summarize the content of the previous chapters. We 
explain the history of the results, give the necessary
references to them, and also discuss the underlying 
motivations. We also present some results which are related 
to the subject of this work only in an indirect way, but 
they give a better insight into it.


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