Péter Major

MULTIPLE WIENER–ITÔ INTEGRALS with applications to limit theorems

– Lecture Note – 849

July 13, 2013

Springer

Contents

1 On a limit problem ............................................. 1

1.1 A brief overview about some results on generalized functions...... 5 2 Wick polynomials .............................................. 9 3 Random spectral measures ...................................... 15

3.1 On the spectral representation of the covariance function of

stationary random fields ..................................... 20

4 Multiple Wiener–Itô integrals ................................... 27 5 The proof of Itô’s formula. The diagram formula and some of its

consequences .................................................. 43 6 Subordinated random fields. Construction of self-similar fields ...... 61 7 On the original Wiener–Itô integral .............................. 75 8 Non-central limit theorems ...................................... 81 9 History of the problems. Comments .............................. 105

References ..................................................... 113 Index ............................................................. 115

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–Itô 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–Itô 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–Itô integrals were worked out at the same time and independently of each other. Actually, we discuss a modified version of the multiple Wiener–Itô 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–Itô integrals and the classical stochastic Itô 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–Itô 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–Itô 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 fields. Consequently we also have to discuss some questions about generalized fields.

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–Itô 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.

Some additional remarks.

This text is a slightly modified version of my Lecture Note Multiple Wiener–Itô integrals with applications to limit theorems published in the Lecture Notes in Mathematics series (number 849) of the Springer Verlag in 1981. I decided to make a special lecture on the basis of this work in the first semester of the university course in 2011–2012 at the University of Szeged. Preparing for it I observed how difficult the reading of formulas in this Lecture Note is. 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. This demanded some changes. It implied e.g. to follow such partly typographical partly linguistic rules by which one does not start a sentence with a formula. Besides, it suggested to formulate the basic definitions in a (typographically) more explicit form and not as an explanation inside the text. When typing this work I also tried to rethink what I had written, to correct the errors and to make the proofs more understandable. It was surprising and a little bit shocking to meet my old personality by studying my old Lecture Note and to recognize how much I have changed. Now I would expose many details in a different way. Naturally I would also make many changes by taking into account the results proved since the time I wrote this note. Nevertheless 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. (There were many such places.) In doing so 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 it, but it has a French version: ‘Le mieux est l’ennemi du bien’.)

I made only 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 how one gets with the help of this theory those results about the so-called spectral representation of the covariance function of stationary random fields that I have formulated under the name Bochner’s theorem and Bochner–Schwartz theorem. This extension of the text is contained in the attachments to Chapter 1 and 3. In the original 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 selfcontained presentation of the subject I have to explain them in more detail.

Budapest, 15 August 2011

Péter Major

Chapter 1

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–Itô integral proved to be a very useful tool at the investigation of this problem.

We shall consider a set of random variables ξ_{n}, n ∈ ℤ_{ν}, where ℤ_{ν} denotes the νdimensional integer lattice, and we shall study their properties. Such a set of random variables will be called a (ν-dimensional) discrete random field. We shall be mainly interested in so-called stationary random fields. Let us recall their definition.

Definition of discrete (strictly) stationary random fields. A set of random variables ξ_{n}, n ∈ ℤ_{ν}, is called a (strictly) stationary discrete random field if

(ξ_{n_{1}},...,ξ_{n_{k}}) =^{Δ} (ξ_{n_{1}+m},...,ξ_{n_{k}+m})

for all k = 1,2,... and n_{1},...,n_{k}, m ∈ ℤ_{ν}, where =^{Δ} denotes equality in distribution.

Let us also recall that a discrete random field ξ_{n}, n ∈ ℤ_{ν}, is called Gaussian if for every finite subset {n_{1},...,n_{k}} ⊂ ℤ_{ν} the random vector (ξ_{n_{1}},...,ξ_{n_{k}}) is normally distributed.

Given a discrete random field ξ_{n}, n ∈ ℤ_{ν}, we define for all N = 1,2,... the new random fields

Z_{n}^{N} = A_{N}^{−1} ∑_{j∈B_{n}^{N}} ξ_{j},  N = 1,2,..., n ∈ ℤ_{ν},  (1.1)

where

B_{n}^{N} = {j: j ∈ ℤ_{ν}, n^{(i)}N ≤ j^{(i)} < (n^{(i)}+ 1)N, i = 1,2,...,ν},

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 distribution of the random fields Z_{n}^{N} defined in (1.1) have a limit as N → ∞. In particular, we would like to describe those random fields Z_{n}^{∗}, n ∈ ℤ_{ν}, which appear as the limit of such random fields Z_{n}^{N}. This problem led to the introduction of the following notion.

Definition of self-similar (discrete) random fields. A (discrete) random field ξ_{n}, n ∈ ℤ_{ν}, is called self-similar with self-similarity parameter α if the random fields Z_{n}^{N} defined in (1.1) with their help and the choice A_{N} = N^{α} satisfy the relation

(ξ_{n_{1}},...,ξ_{n_{k}}) =^{Δ} (Z_{n_{1}}^{N},...,Z_{n_{k}}^{N})  (1.2)

for all N = 1,2,... and n_{1},...,n_{k} ∈ ℤ_{ν}.

We are interested in the choice A_{N} = N^{α} with some α > 0 in the definition of the random variables Z_{n}^{N} in (1.2), because under slight restrictions, relation (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 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^{ℤ^{ν}} induced by formula (1.1) have a very complicated structure. We shall define the so-called subordinated fields below. (More precisely the fields subordinated to a stationary Gaussian field.) In case of subordinated fields the Wiener–Itô integral is a very useful tool for investigating the transformation defined in (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 ∈ ℤ_{ν}, be a stationary Gaussian field. We define the shift transformations T_{m}, m ∈ ℤ_{ν}, over this field by the formula T_{m}X_{n} = X_{n+m} for all n, m ∈ ℤ_{ν}. Let H denote the real Hilbert space consisting of the square integrable random variables measurable with respect to the σ-algebra B = B(X_{n}, n ∈ ℤ_{ν}). The scalar product in H is defined as (ξ,η) = Eξη, ξ, η ∈ H . The shift transformations T_{m}, m ∈ ℤ_{ν}, can be extended to a group of unitary shift transformations over H in a natural way. Namely, if ξ = f(X_{n_{1}},...,X_{n_{k}}) then we define T_{m}ξ = f(X_{n_{1}+m},...,X_{n_{k}+m}). It can be seen that ∥ξ∥ = ∥T_{m}ξ∥, and the above considered random variables ξ are dense in H . (A more detailed discussion about the definition of shift operators and their properties will be given in Chapter 2 in a Remark after the formulation of Theorem 2C. Here we shall define the shift T_{m}ξ, m ∈ ℤ_{ν}, of all random variables ξ which are measurable with respect to the σ-algebra B(X_{n}, n ∈ Z_{ν}), i.e. ξ does not have to be square integrable.) Hence ∥T_{m}∥ can be extended to the whole space H by L_{2} continuity. It can be proved that the norm preserving transformations T_{m}, m ∈ ℤ_{ν}, constitute a unitary group in H , i.e. T_{n+m} = T_{n}T_{m} for all n, m ∈ ℤ_{ν}, and T_{0} = Id. Now we introduce the following

Definition of subordinated random fields. Given a stationary Gaussian field X_{n}, n ∈ ℤ_{ν}, we define the Hilbert spaces H and the shift transformations T_{m}, m ∈ ℤ_{ν}, over H as before. A discrete stationary field ξ_{n} is called a random field subordinated to X_{n} if ξ_{n} ∈ H , and T_{n}ξ_{m}= ξ_{n+m} for all n, m ∈ ℤ_{ν}.

We remark that ξ_{0} determines the subordinated fields ξ_{n} completely, since ξ_{n} = T_{n}ξ_{0}. Later we give a more adequate description of subordinated fields by means of Wiener–Itô 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 φ_{n}(x), n ∈ ℤ_{ν}, n = (n_{1},...,n_{ν}), denote the indicator function of the cube [n_{1}− \frac{1}{2},n_{1}+ \frac{1}{2})×··· ×[n_{ν} − \frac{1}{2},n_{ν} + \frac{1}{2}), with center n = (n_{1},...,n_{ν}) and with edges of length 1, i.e. let φ_{n}(x) = 1, x = (x_{1},...,x_{ν}) ∈ R^{ν}, if n_{j} − \frac{1}{2} ≤ x_{j} < n_{j} + \frac{1}{2} for all 1 ≤ j ≤ ν, and let φ_{n}(x) = 0 otherwise. Define the linear space Φ of functions on R^{ν} consisting of all finite linear combinations of the form ∑ c_{j}φ_{n_{j}}(x), n_{j} ∈ ℤ_{ν}, with the above defined functions φ_{n}(x) and real coefficients c_{j}. Given a discrete random field ξ_{n}, n ∈ ℤ_{ν}, define the random variables ξ(φ) for all φ ∈ Φ in the following way. Put ξ(φ) = ∑ c_{j}ξ_{n_{j}} if φ(x) = ∑ c_{j}φ_{n_{j}}(x). In particular, ξ(φ_{n}) = ξ_{n} for all n ∈ ℤ_{ν}. The identity ξ(c_{1}φ + c_{2}ψ) = c_{1}ξ(φ) + c_{2}ξ(ψ) also holds for all φ,ψ ∈ Φ and real numbers c_{1} and c_{2}.

Let us also define the function φ^{(N,A_{N})}(x) = \frac{1}{A_{N}}φ(\frac{x}{N}) for all functions φ ∈ Φ and positive integers N = 1,2,..., with some appropriately chosen constants A_{N} > 0. Observe that ξ(φ_{n}^{(N,A_{N})}) = Z_{n}^{N} with the random variable Z_{n}^{N} defined in (1.1). All previously introduced notions related to discrete random fields can be reformulated with the help of the set of random variables ξ(φ), φ ∈ Φ. Thus for instance the random field ξ_{n}, n ∈ ℤ_{ν} is self-similar with self-similarity parameter α if and only if ξ(φ^{(N,N^{α})}) =^{Δ} ξ(φ) for all φ ∈ Φ and N = 1,2,.... (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 ξ(φ), φ ∈ Φ, defined above. The main difference is that instead of the space Φ 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 S = S_{ν} be the Schwartz space of (real valued) rapidly decreasing, smooth functions on R^{ν}. (See e.g. [15] for the definition of S_{ν}. I shall present a more detailed discussion about the definition of the space S in the adjustment to Chapter 1.) Generally one takes the space of complex valued, rapidly decreasing, smooth functions as the space S, but we shall denote the space of real valued, rapidly decreasing, smooth functions by S if we do not say this otherwise. We shall omit the subscript ν if it leads to no ambiguity. Now we introduce the notion of generalized random fields.

Definition of generalized random fields. We say that the set of random variables X(φ), φ ∈ S, is a generalized random field over the Schwartz space S of rapidly decreasing, smooth functions if:

(a) X(a_{1}φ_{1}+a_{2}φ_{2}) = a_{1}X(φ_{1})+a_{2}X(φ_{2}) with probability 1 for all real numbers a_{1} and a_{2} and φ_{1} ∈ S, φ_{2} ∈ S. (The exceptional set of probability 0 where this identity does not hold may depend on a_{1}, a_{2}, φ_{1} and φ_{2}.)

(b) X(φ_{n}) ⇒ X(φ) stochastically if φ_{n} → φ in the topology of S.

We also introduce the following definitions.

Definition of stationarity and Gaussian property of a generalized random field. On the notion of convergence of generalized random fields in distribution. The generalized random field X = {X(φ), φ ∈ S } is stationary if X(φ) =^{Δ} X(T_{t}φ) for all φ ∈ S and t ∈ R^{ν}, where T_{t}φ(x) = φ(x−t). It is Gaussian if X(φ) is a Gaussian random variable for all φ ∈ S. The relation X_{n} →^{D} X_{0} as n → ∞ holds for a sequence of generalized random fields X_{n}, n = 0,1,2,..., if X_{n}(φ) →^{D} X_{0}(φ) for all φ ∈ S, where →^{D} denotes convergence in distribution.

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_{t}^{A} for all t > 0 by the formula

X_{t}^{A}(φ) = X(φ_{t}^{A}), φ ∈ S ,  where φ_{t}^{A}(x) = A(t)^{−1} φ ( \frac{x}{t}) .  (1.3)

We are interested in the following

Question. When does a generalized random field X^{∗} exist such that X_{t}^{A} →^{D} X^{∗} as t → ∞ (or as t → 0)?

In relation to this question we introduce the following

Definition of self-similarity. The stationary generalized random field X is selfsimilar with self-similarity parameter α if X_{t}^{A}(φ) =^{Δ} X(φ) for all φ ∈ S and t > 0 with the function A(t) = t^{α}.

To answer the above question one should first describe the generalized selfsimilar random fields.

We try to explain the motivation behind the above definitions. Given an ordinary random field X(t), t ∈ R^{ν}, and a topological space E consisting of functions over R^{ν} one can define the random variables X(φ) = ∫_{R^{ν}}φ(t)X(t)dt, φ ∈ E. Some difficulty may arise when defining this integral, but it can be overcome in all interesting cases. If the space E is rich enough, and this is the case if E = S, then the integrals X(φ), φ ∈ E, determine the random process X(t). The set of random variables X(φ), φ ∈ S, 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(φ), φ ∈ S, and we can extend the space S containing the test function φ to such a larger linear space T for which Φ ⊂ T with the above introduced linear space Φ, then we can define the discrete random field X(φ), φ ∈ Φ, by a restriction of the space of test functions of the generalized random field X(φ), φ ∈ T . This random field can be considered as the discretization of the original generalized random field X(φ), φ ∈ S.

We finish this chapter by defining the generalized subordinated random fields. The we shall explain the basic results about the Schwartz space S and generalized functions in a separate sub chapter.

Let X(φ), φ ∈ S, be a generalized stationary Gaussian random field. The formula T_{t}X(φ)) = X(T_{t}φ), T_{t}φ(x) = φ(x − t), defines the shift transformation for all t ∈ R^{ν}. Let H denote the real Hilbert space consisting of the B = B(X(φ), φ ∈ S) measurable random variables with finite second moment. The shift transformation can be extended to a group of unitary transformations over H similarly to the discrete case.

Definition of generalized random fields subordinated to a generalized stationary Gaussian random field. Given a generalized stationary Gaussian random field X(φ), φ ∈ S, we define the Hilbert space H and the shift transformations T_{t}, t ∈ R^{ν}, over H as above. A generalized stationary random field ξ(φ), φ ∈ S, is subordinated to the field X(φ), φ ∈ S, if ξ(φ) ∈ H and T_{t}ξ(φ) = ξ(T_{t}φ) for all φ ∈ S and t ∈ R^{ν}, and E[ξφ_{n}) − ξ(φ)]^{2} → 0 if φ_{n} → φ in the topology of S.

1.1 A brief overview about some results on generalized functions

Let us first describe the Schwartz spaces S and S^{c} in more detail. The space S^{c} = (S_{ν})^{c} consists of those complex valued functions of ν variables which decrease at infinity, together with their derivatives, faster than any polynomial degree. More explicitly, φ ∈ S^{c} for a complex valued function φ of ν variables if

|x_{1}^{k_{1}}··· x_{ν}^{k_{ν}} \frac{∂^{q_{1}+··· +q_{ν}}}{∂x_{1}^{q_{1}}...∂x_{ν}^{q_{ν}}}φ(x_{1},...,x_{ν})|≤ C(k_{1},...,k_{ν},q_{1},...,q_{ν})

for all point x = (x_{1},...,x_{ν}) ∈ R^{ν} and vectors (k_{1},...,k_{ν}), (q_{1},...,q_{ν}) with nonnegative integer coordinates with some constant C(k_{1},...,k_{ν},q_{1},...,q_{ν}) which may depend on the function φ. This formula can be written in a more concise form as

|x^{k}D^{q}φ(x)| ≤ C(k,q) with k = (k_{1},...,k_{ν}) and q = (q_{1},...,q_{ν}),

where x = (x_{1},...,x_{ν}), x^{k} = x_{1}^{k_{1}}··· x_{ν}^{k_{ν}} and D^{q} = \frac{∂^{q_{1}+··· +qν}}{∂x_{1}^{q_{1}}...∂x_{ν}^{qν}}. The elements of the

space S are defined similarly, with the only difference that they are real valued functions.

To define the spaces S and S^{c} we still have to define the convergence in them. We say that a sequence of functions φ_{n} ∈ S^{c} (or φ_{n} ∈ S) converges to a function φ if

lim_{n→∞} sup_{x∈R^{ν}}(1 + |x|^{2})^{k}|D^{q}φ_{n}(x) − D^{q}φ(x)| = 0.

for all k = 1,2,... and q = (q_{1},...,q_{ν}). It can be seen that the limit function φ is also in the space S^{c} (or in the space S).

A nice topology can be introduced in the space S^{c} (or S) which induces the above convergence. The following topology is an appropriate choice. Let a basis of neighbourhoods of the origin consist of the sets

U(k,q,ε) = {φ: max_{x}(1 + |x|^{2})^{k}|D^{q}φ(x)| < ε}

with k = 0,1,2,..., q = (q_{1},...,q_{ν}) with non-negative integer coordinates and ε > 0, where |x|^{2} = x_{1}^{2} + ···+ x_{ν}^{2}. A basis of neighbourhoods of an arbitrary function φ ∈ S^{c} (or φ ∈ S) consists of sets of the form φ +U(k,q,ε), where the class of sets U(k,q,ε) is a basis of neighbourhood of the origin. The fact that the convergence in 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 S′ consists of the continuous linear maps F : S → C or F : S^{c} → C, where C denotes the linear space of complex numbers. (In the study of the space 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 (φ), F ∈ S′and φ ∈ S (or φ ∈ S^{c}) in the form (F,φ).

We can define generalized functions F ∈ S′by the formula

(F,φ) = ∫ \overline{f(x)}φ(x)dx for all φ ∈ S or φ ∈ S^{c}

with a function f such that ∫(1+|x|^{2})^{−p} |f(x)|dx < ∞ with some p ≥ 0. (The upper script ¯ denotes complex conjugate in the sequel.) Such functionals are called regular. There are also non-regular functionals in the space S′. An example for them is the δ-function defined by the formula (δ,φ) = φ(0). There is a good description of the generalized functions F ∈ 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 that we omit is about the interpretation of a usual function as a generalized function in the case when it does not define a regular function in 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 find 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{∂_{j}}{∂x_{j}}F,φ) = −(F, \frac{∂φ}{∂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

∫_{R^{ν}} \overline{f(x)}g(x)dx = \frac{1}{(2π)^{ν}} ∫_{R^{ν}} \overline{f̃(u)}g̃(u)du

is formulated with f̃(u) = ∫_{R^{ν}}e^{i(u,x)}f(x)dx and g̃(u) = ∫_{R^{ν}}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 ∈ 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 ∈ S^{c} can be calculated as the inverse Fourier transform of its Fourier transform g̃∈ S^{c}, i.e. g(x) = \frac{1}{(2π)^{ν}} ∫ e^{−i(u,x)} g̃(u)du, we can write

∫ \overline{f(x)}g(x)dx= ∫ \overline{f(x)}[ \frac{1}{(2π)^{ν}} ∫ e^{−i(u,x)} g̃(u)du]dx
= ∫ g̃(u)[ \frac{1}{(2π)^{ν}} ∫ \overline{e^{i(u,x)}f(x)}dx] du
= \frac{1}{(2π)^{ν}} ∫ \overline{f̃(u)}g̃(u)du.

Let us also remark that the Fourier transform f → f̃is a bicontinuous map from S^{c} to S^{c}. (This means that this transformation is invertible, and both the Fourier transform and its inverse are continuous maps from S^{c} to S^{c}.) (The restriction of the Fourier transform to the space S of real valued functions is a bicontinuous map from S to the subspace of S^{c} consisting of those functions f ∈ S^{c} for which f(−x) = \overline{f(x)} for all x ∈ R^{ν}.)

The above results make natural the following definition of the Fourier transform F̃ of a generalized function F ∈ S′.

(F̃,φ̃) = (2π)^{ν}(F,φ) for all φ ∈ S^{c}.

Indeed, if F ∈ S′then F̃ is also a continuous linear map on S^{c}, i.e. it is also an element of S′. Besides, the above proved version of the Parseval formula implies that if we consider an integrable function f on R^{ν} both as a usual function and as a (regular) generalized function, its Fourier transform agrees in the two cases.

There are other classes of test functions and spaces of generalized functions studied in the literature. The most popular among them is the space D of infinitely many differentiable functions with compact support and its dual space D′, the space of continuous linear transformations on the space D. (These spaces are generally denoted by D and D′in the literature, although just the book [15] that we use as our main reference in this subject applies the notation K and K ′for them.) We shall discuss this space only very briefly.

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

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

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– Itô integrals.

Let X_{t}, t ∈ T, be a set of jointly Gaussian random variables indexed by a parameter set T. Let EX_{t} = 0 for all t ∈ T. We define the real Hilbert spaces H_{1} and H in the following way: A square integrable random variable is in H if and only if it is measurable with respect to the σ-algebra B = B(X_{t}, t ∈ T), and the scalar product in H is defined as (ξ,η) = Eξη, ξ, η ∈ H . The Hilbert space H_{1} ⊂ H is the subspace of H generated by the finite linear combinations ∑ c_{j}X_{t_{j}}, t_{j} ∈ T. We consider only such sets of Gaussian random variables X_{t} for which H_{1} is separable. Otherwise X_{t}, t ∈ T, can be arbitrary, but the most interesting case for us is when T = S_{ν} or ℤ_{ν}, and X_{t}, t ∈ T, is a stationary Gaussian field.

Let Y_{1},Y_{2},... be an orthonormal basis in H_{1}. The uncorrelated random variables Y_{1},Y_{2},... are independent, since they are (jointly) Gaussian. Moreover,

B(Y_{1},Y_{2},...) = B(X_{t}, t ∈ T).

Let H_{n}(x) denote the n-th Hermite polynomial with leading coefficient 1, i.e. let H_{n}(x) = (−1)^{n}e^{x^{2}/2} \frac{d^{n}}{dx^{n}}(e^{−x^{2}/2}). We recall the following results from analysis and measure theory.

Theorem 2A. The Hermite polynomials H_{n}(x), n = 0,1,2,..., form a complete orthogonal system in L_{2}( R,B, \frac{1}{\sqrt{2π}} e^{−x^{2}/2} dx). (Here B denotes the Borel σ-algebra on the real line.)

Let (X_{j},X_{j},μ_{j}), j = 1,2,..., be countably many independent copies of a probability space (X,X ,μ). (We denote the points of X_{j} by x_{j}.) Let (X^{∞} ,X^{∞} ,μ^{∞}) =

∏_{j=1}^{∞} (X_{j},X_{j},μ_{j}). With such a notation the following result holds.

Theorem 2B. Let φ_{0},φ_{1},..., φ_{0}(x) ≡ 1, be a complete orthonormal system in the

Hilbert space L_{2}(X,X ,μ). Then the functions ∏_{j=1}^{∞} φ_{k_{j}}(x_{j}), where only finitely many

indices k_{j} differ from 0, form a complete orthonormal basis in L_{2}(X^{∞} ,X^{∞} ,μ^{∞}).

Theorem 2C. Let Y_{1},Y_{2},... be random variables on a probability space (Ω,A ,P) taking values in a measurable space (X,X ). Let ξ be a real valued random variable measurable with respect to the σ-algebra B(Y_{1},Y_{2},...), and let (X^{∞} ,X^{∞}) denote the infinite product (X × X × ···,X × X × ··· ) of the space (X,X ) with itself. Then there exists a real valued, measurable function f on the space (X^{∞} ,X^{∞}) such that ξ = f(Y_{1},Y_{2},...).

Remark. Let us have a stationary random field X_{n}(ω), n ∈ ℤ_{ν}. Theorem 2C enables us to extend the shift transformation T_{m}, defined as T_{m}X_{n}(ω) = X_{n+m}(ω), n, m ∈ ℤ_{ν}, for all random variables ξ(ω), measurable with respect to the σ-algebra B(X_{n}(ω), n ∈ ℤ_{ν}). Indeed, by Theorem 2C we can write ξ(ω) = f(X_{n}(ω), n ∈ ℤ_{ν}), and define T_{m}ξ(ω) = f(X_{n+m}(ω), n ∈ ℤ_{ν}). We still have to understand, that although the function f is not unique in the representation of the random variable ξ(ω), the above definition of T_{m}ξ(ω) is meaningful. To see this we have to observe that if f_{1}(X_{n}(ω), n ∈ ℤ_{ν}) = f_{2}(X_{n}(ω), n ∈ ℤ_{ν}) for two functions f_{1} and f_{2} with probability 1, then also f_{1}(X_{n+m}(ω), n ∈ ℤ_{ν}) = f_{2}(X_{n+m}(ω), n ∈ ℤ_{ν}) with probability 1 because of the stationarity of the random field X_{n}(ω), n ∈ ℤ_{ν}. Let us also observe that ξ(ω) =^{Δ} T_{m}ξ(ω) for all m ∈ ℤ_{ν}. Besides, T_{m} is a linear operator on the linear space of random variables, measurable with respect to the σ-algebra B(X_{n}, n ∈ ℤ_{ν}). 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 ∈ ℤ_{ν}, constitute a unitary group.

Let a stationary generalized field X = {X(φ), φ ∈ S } be given. The shift T_{t}ξ of a random variable ξ, measurable with respect to the σ-algebra B(X(φ), φ ∈ S) can be defined for all t ∈ R^{ν} similarly to the discrete case with the help of Theorem 2C and the following result. If ξ ∈ B(X(φ), φ ∈ S) for a random variable ξ, then there exists such a countable subset {φ_{1},φ_{2},...} ⊂ S (depending on the random variable ξ) for which ξ is B(X(φ_{1}),X(φ_{2}),...) measurable. (We write ξ(ω) = f(X(φ_{1})(ω),X(φ_{2})(ω),...) with appropriate functions f, and φ_{1} ∈ S, φ_{2} ∈ S,…, and define the shift T_{t}ξ as T_{t}ξ(ω) = f(X(T_{t}φ_{1})(ω),X(T_{t}φ_{2})(ω),...), where T_{t}φ(x) = φ(x − t) for φ ∈ S.) The transformations T_{t}, t ∈ R^{ν}, are linear operators over the space of random variables measurable with respect to the σ-algebra B(X(φ), φ ∈ S) with similar properties as their discrete counterpart.

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

Theorem 2.1. Let Y_{1},Y_{2},... be an orthonormal basis in the Hilbert space H_{1} defined above with the help of a set of Gaussian random variables X_{t}, t ∈ T. Then the set of all possible finite products H_{j_{1}}(Y_{l_{1}})··· H_{j_{k}}(Y_{l_{k}}) is a complete orthogonal system in the Hilbert space H defined above. (Here H_{j}(·) denotes the j-th Hermite polynomial.)

Proof of Theorem 2.1. By Theorems 2A and 2B the set of all possible prod-

ucts ∏_{j=1}^{∞} H_{k_{j}}(x_{j}), where only finitely many indices k_{j} differ from 0, is a com-

plete orthonormal system in L_{2}( R^{∞} ,B^{∞} , ∏_{j=1}^{∞} \frac{e^{−x^{2}_{j}/2}}{\sqrt{2π}} dx_{j}) . Since B(X_{t}, t ∈ T) =

B(Y_{1},Y_{2},...), Theorem 2C implies that the mapping f(x_{1},x_{2},...,) → f(Y_{1},Y_{2},...)

is a unitary transformation from L_{2}( R^{∞} ,B^{∞} , ∏_{j=1}^{∞} \frac{e^{−x^{2}_{j}/2}}{\sqrt{2π}} dx_{j}) to 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. ⊓ ⊔

Let H_{≤n} ⊂ H , n = 1,2,..., (with the previously introduced Hilbert space H ) denote the Hilbert space which is the closure of the linear space consisting of the elements P_{n}(X_{t_{1}},...,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},...,t_{m} ∈ T are arbitrary. Let H_{0} = H_{≤0} consist of the constant functions, and let H_{n} = H_{≤n} ⊖ H_{≤n−1} , n = 1,2,..., where ⊖ denotes orthogonal completion. It is clear that the Hilbert space H_{1} given in this definition agrees with the previously defined Hilbert space H_{1}. If ξ_{1},...,ξ_{m}∈ H_{1}, and P_{n}(x_{1},...,x_{m}) is a polynomial of degree n, then P_{n}(ξ_{1},...,ξ_{m}) ∈ H_{≤n}. Hence Theorem 2.1 implies that

H = H_{0}+ H_{1}+ H_{2}+ ···,  (2.1)

where + denotes direct sum. Now we introduce the following

Definition of Wick polynomials. Given a polynomial P(x_{1},...,x_{m}) of degree n and a set of (jointly Gaussian) random variables ξ_{1},...,ξ_{m} ∈ H_{1}, the Wick polynomial : P(ξ_{1},...,ξ_{m}): is the orthogonal projection of the random variable P(ξ_{1},...,ξ_{m}) to the above defined subspace H_{n} of the Hilbert space H .

It is clear that Wick polynomials of different degree are orthogonal. Given some ξ_{1},...,ξ_{m} ∈ H_{1} define the subspaces H_{≤n} (ξ_{1},...,ξ_{m}) ⊂ H_{≤n}, n = 1,2,..., as the set of all polynomials of the random variables ξ_{1},...,ξ_{m} with degree less than or equal to n. Let H_{≤0} (ξ_{1},...,ξ_{m}) = H_{0}(ξ_{1},...,ξ_{m}) = H_{0}, and H_{n}(ξ_{1},...,ξ_{m}) = H_{≤n} (ξ_{1},...,ξ_{m}) ⊖ H_{≤n−1} (ξ_{1},...,ξ_{m}). With the help of this notation we formulate the following

Proposition 2.2. Let P(x_{1},...,x_{m}) be a polynomial of degree n. Then the random polynomial : P(ξ_{1},...,ξ_{m}): equals the orthogonal projection of P(ξ_{1},...,ξ_{m}) to H_{n}(ξ_{1},...,ξ_{m}).

Proof of Proposition 2.2. Let : P̄(ξ_{1},...,ξ_{m}): denote the projection of the random polynomial P(ξ_{1},...,ξ_{m}) to H_{n}(ξ_{1},...,ξ_{m}). Obviously

P(ξ_{1},...,ξ_{m})−: P̄(ξ_{1},...,ξ_{m}): ∈ H_{≤n−1} (ξ_{1},...,ξ_{m}) ⊆ H_{≤n−1} .

Hence in order to prove Proposition 2.2 it is enough to show that for all η ∈ H_{≤n−1}

E: P̄(ξ_{1},...,ξ_{m}): η = 0,  (2.2)

since this means that : P̄(ξ_{1},...,ξ_{m}): is the orthogonal projection of P(ξ_{1},...,ξ_{m}) ∈ H_{≤n} to H_{≤n−1} .

Let ε_{1},ε_{2},... be an orthonormal system in H_{1}, also orthonormal to ξ_{1},...,ξ_{m},

and such that ξ_{1},...,ξ_{m},ε_{1},ε_{2},... form a basis in H_{1}. If η = ∏_{i=1}^{m} ξ_{i}^{l_{i}} ∏_{j=1}^{∞} ε_{j}^{k_{j}} with such

exponents l_{i} and k_{j} that ∑ l_{i}+∑ k_{j} ≤ n−1, then (2.2) holds for this random variable η because of the independence of the random variables ξ_{i} and ε_{j}. Since the linear combinations of such η are dense in H_{≤n−1} , formula (2.2) and Proposition (2.2) are proved. ⊓ ⊔

Corollary 2.3. Let ξ_{1},...,ξ_{m} be an orthonormal system in H_{1}, and let

P(x_{1},...,x_{m}) = ∑ c_{j_{1},...,j_{m}} x^{j_{1}} ··· x_{m}^{j_{m}}

be a homogeneous polynomial, i.e. let j_{1}+ ··· j_{m}= n with some fixed number n for all sets (j_{1},...,j_{m}) appearing in this summation. Then

: P(ξ_{1},...,ξ_{m}): = ∑ c_{j_{1},...,j_{m}} H_{j_{1}}(ξ_{1})··· H_{j_{m}}(ξ_{m}).

In particular,

: ξ^{n}: = H_{n}(ξ) ifξ ∈ H_{1}, and Eξ^{2} = 1.

Remark. Although we have defined the Wick polynomial (of degree n) for all polynomials P(ξ_{1},...,ξ_{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},...j_{m})ξ_{1}^{l_{1}} ··· ξ_{m}^{l_{m}} of the polynomial P(ξ_{1},...,ξ_{m}) such that l_{1} + ···+ l_{m} < n has a zero contribution in the definition of the Wick polynomial : P(ξ_{1},...,ξ_{m}): .

Proof of Corollary 2.3. Let the degree of the polynomial P be n. Then

P(ξ_{1},...,ξ_{m}) − ∑ c_{j_{1},...,j_{m}} H_{j_{1}}(ξ_{1})··· H_{j_{m}}(ξ_{m}) ∈ H_{≤n−1} (ξ_{1},...,ξ_{m}),  (2.3)

since P(ξ_{1},...,x_{m}) − ∑ c_{j_{1},...,j_{m}} H_{j_{1}}(ξ_{1})··· H_{j_{m}}(ξ_{m}) is a polynomial whose degree is

less than n. Let η = ξ_{1}^{l_{1}} ··· ξ_{m}^{l_{m}}, ∑_{i=1}^{m} l_{i} ≤ n − 1. Then

EηH_{j_{1}}(ξ_{1})··· H_{j_{m}}(ξ_{m}) = ∏_{i=1}^{m} Eξ_{i}^{l_{i}}H_{j_{i}}(ξ_{i}) = 0,

since l_{i} < j_{i} for at least one index i. Therefore

Eη ∑ c_{j_{1},...,j_{m}} H_{j_{1}}(ξ_{1})··· H_{j_{m}}(ξ_{m}) = 0.  (2.4)

Since every element of H_{≤n−1} (ξ_{1},...,ξ_{m}) can be written as the sum of such elements η, relation (2.4) holds for all η ∈ H_{≤n−1} (ξ_{1},...,ξ_{m}). Relations (2.3) and (2.4) imply Corollary 2.3. ⊓ ⊔

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

Corollary 2.4. Let ξ_{1},ξ_{2},... be an orthonormal basis in H_{1}. Then the random variables H_{j_{1}}(ξ_{1})··· H_{j_{k}}(ξ_{k}), k = 1,2,..., j_{1}+··· +j_{k}= n, form a complete orthogonal basis in H_{n}.

Proof of Corollary 2.4. It follows from Corollary 2.3 that

H_{j_{1}}(ξ_{1})··· H_{j_{k}}(ξ_{k}) =: ξ_{1}^{j_{1}} ··· ξ_{k}^{j_{k}}: ∈ H_{n} for all k = 1,2,...

if j_{1} + ··· + j_{k} = n. These random variables are orthogonal, and all Wick polynomials : P(ξ_{1},...,ξ_{m}): of degree n of the random variables ξ_{1},ξ_{2},... can be represented as the linear combination of such terms. Since these Wick polynomials are dense in H_{n}, this implies Corollary 2.4. ⊓ ⊔

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 (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 H_{1}. It can be extended to a unitary transformation U over H in a natural way. Fix an orthonormal basis ξ_{1},ξ_{2},... in H_{1}, and define U 1 = 1, U ξ_{j_{1}}^{l_{1}} ··· ξ_{j_{k}}^{l_{k}} = (Uξ_{j_{1}})^{l_{1}} ··· (Uξ_{j_{k}})^{l_{k}}. This transformation can be extended to a linear transformation U over H in a unique way. The transformation U is norm preserving, since the joint distributions of (ξ_{j_{1}},ξ_{j_{2}},...) and (Uξ_{j_{1}},Uξ_{j_{2}},...) coincide. Moreover, it is unitary, since Uξ_{1},Uξ_{2},... is an orthonormal basis in H_{1}. It is not difficult to see that if P(x_{1},...,x_{m}) is an arbitrary polynomial, and η_{1},η_{2}...,η_{m} ∈ H_{1}, then U P(η_{1},...,η_{m}) = P(Uη_{1},...,Uη_{m}). This relation means in particular that the transformation U does not depend on the choice of the basis in H_{1}. If the transformations U_{1} and U_{2} correspond to two unitary transformations U_{1} and U_{2} on H_{1}, then the transformation U_{1}U_{2} corresponds to U_{1}U_{2}. The subspaces H_{≤n} and therefore the subspaces H_{n} remain invariant under the transformations U .

The shift transformations of a stationary Gaussian field, and their extensions to H are the most interesting examples for such unitary transformations U and U . In the terminology of group representations the above facts can be formulated in the following way: The mapping U → U is a group representation of U(H_{1}) over H , where U(H_{1}) denotes the group of unitary transformations over H_{1}. Formula (2.1) gives a decomposition of H into orthogonal invariant subspaces of this representation.

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 socalled 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–Itô 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 ∈ ℤ_{ν}, or X(φ), φ ∈ S, we shall assume throughout the paper that EX_{n} = 0, EX_{n}^{2} = 1 in the discrete and EX(φ) = 0 in the generalized field case.

Theorem 3A. (Bochner.) Let X_{n}, n ∈ ℤ_{ν}, be a discrete (Gaussian) stationary field. There exists a unique probability measure G on [−π,π)^{ν} such that the correlation function r(n) = EX_{0}X_{n} = EX_{k}X_{k+n}, n ∈ ℤ_{ν}, k ∈ ℤ_{ν}, can be written in the form

r(n) = ∫ e^{i(n,x)}G(dx),  (3.1)

where (·,·) denotes scalar product. Further G(A) = G(−A) for all A ∈ [−π,π)^{ν}.

We can identify [−π,π)^{ν} with the torus R^{ν}/2πℤ_{ν}. Thus e.g. −(−π,...,−π) = (−π,...,−π).

Theorem 3B. (Bochner–Schwartz.) Let X(φ), φ ∈ S, be a generalized Gaussian stationary random field over S = S_{ν}. There exists a unique σ-finite measure G on R^{ν} such that

EX(φ)X(ψ) = ∫ φ̃(x) ¯ψ̃(x)G(dx)  for all φ, ψ ∈ S ,  (3.2)

where ˜ denotes Fourier transform and ¯ complex conjugate. The measure G has the properties G(A) = G(−A) for all A ∈ B^{ν}, and

∫ (1 + |x|)^{−r} G(dx) < ∞ with an appropriate r > 0.  (3.3)

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 formulated results follow from them.

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.

Let us now consider a stationary Gaussian random field (discrete or generalized one) with spectral measure G. We shall denote the space L_{2}([−π,π)^{ν},B^{ν},G) or L_{2}(R^{ν},B^{ν},G) simply by L_{G}^{2}. Let H_{1} be the real Hilbert space defined by means of the stationary random field, as it was done in Chapter 2. Let H_{1}^{c} denote its complexification, i.e. the elements of H_{1}^{c} are of the form X +iY, X, Y ∈ H_{1}, and the scalar product is defined as (X_{1}+iY_{1},X_{2}+iY_{2}) = EX_{1}X_{2}+EY_{1}Y_{2}+i(EY_{1}X_{2}−EX_{1}Y_{2}). We are going to construct a unitary transformation I from L_{G}^{2} to H_{1}^{c}. We shall define the random spectral measure via this transformation.

Let S^{c} denote the Schwartz space of rapidly decreasing, smooth, complex valued functions with the usual topology of the Schwartz space. (The elements of S^{c} are of the form φ + iψ, φ, ψ ∈ S.) We make the following observation. The finite linear combinations ∑ c_{n}e^{i(n,x)}are dense in L_{G}^{2} in the discrete field, and the functions φ ∈ 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 [−π,π)^{ν} 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_{G}^{2}. We claim that also the functions of the space D are dense in L_{G}^{2}, where D denotes the class of (complex valued) infinitely many times differentiable functions with compact support. Indeed, if φ ∈ D is real valued, φ(x) ≥ 0 for all x ∈ R^{ν}, ∫ φ(x)dx = 1, we define φ_{t}(x) = t^{ν}φ ( \frac{x}{t}), and f is a continuous function with compact support, then f ∗ φ_{t} → f uniformly as t → ∞. Here ∗ denotes convolution. On the other hand, f ∗ φ_{t} ∈ D for all t > 0. Hence D ⊂ S^{c} is dense in L_{G}^{2}.

Finally we recall the following result from the theory of distributions. The mapping φ → φ̃ is an invertible, bicontinuous transformation from S^{c} into S^{c}. In particular, the set of functions φ̃, φ ∈ S, is also dense in L_{G}^{2}.

Now we define the mapping

I ( ∑ c_{n}e^{i(n,x)}) = ∑ c_{n}X_{n}  (3.4)

in the discrete case, where the sum is finite, and

I({ \atop φ + iψ)} = X(φ) + iX(ψ), φ, ψ ∈ S  (3.5)

in the generalized case.

Obviously,

∥∑ c_{n}e^{i(n,x)}∥_{L_{G}^{2}}^{2} = ∑ ∑ c_{n}c̄_{m}∫ e^{i(n−m),x}G(dx)
= ∑ ∑ c_{n}c̄_{m}EX_{n}X_{m}= E |∑ c_{n}X_{n}|^{2},

and

∥{ \atop φ + iψ} ∥_{L_{G}^{2}}^{2} = ∫ [φ̃(x) ¯φ̃(x) − iφ̃(x) ¯ψ̃(x) + iψ̃(x) ¯φ̃(x) + ψ̃(x) ¯ψ̃(x)]G(dx)
= EX(φ)^{2}− iEX(φ)X(ψ) + iEX(ψ)X(φ) + EX(ψ)^{2}
= E (|X(φ) + iX(ψ)|)^{2}.

This means that the mapping I from a linear subspace of L_{G}^{2} to H_{1}^{c} is norm preserving. Besides, the subspace where I was defined is dense in L_{G}^{2}, since the space of continuous functions is dense in L_{G}^{2} if G is a finite measure on the torus R^{ν}/2πℤ_{ν}, and the space of continuous functions with a compact support is dense in L_{G}^{2}(R^{ν}) if the measure G satisfies relation (3.3). Hence the mapping I can be uniquely extended to a norm preserving transformation from L_{G}^{2} to H_{1}^{c}. Since the random variables X_{n} or X(φ) are obtained as the image of some element from L_{G}^{2} under this transformation, I is a unitary transformation from L_{G}^{2} to H_{1}^{c}. A unitary transformation preserves not only the norm, but also the scalar product. Hence ∫ f(x)ḡ(x)G(dx) = EI(f)\overline{I(g)}for all f, g ∈ L_{G}^{2}.

Now we define the random spectral measure Z_{G}(A) for all A ∈ B^{ν} such that G(A) < ∞ by the formula

Z_{G}(A) = I(χ_{A}),

where χ_{A}denotes the indicator function of the set A. It is clear that

(i) The random variables Z_{G}(A) are complex valued, jointly Gaussian random variables. (The random variables ReZ_{G}(A) and ImZ_{G}(A) with possibly different sets A are jointly Gaussian.)

(ii) EZ_{G}(A) = 0, (iii) EZ_{G}(A)\overline{Z_{G}(B)}= G(A ∩ B),

(iv) ∑_{j=1}^{n} Z_{G}(A_{j}) = Z_{G}( ⋃_{j=1}^{n} A_{j}) if A_{1},...,A_{n} are disjoint sets.

Also the following relation holds.

(v) Z_{G}(A) = \overline{Z_{G}(−A)}.

This follows from the relation (v′) I(f) = \overline{I(f_{−} )} for all f ∈ L_{G}^{2}, where f_{−} (x) = \overline{f(−x)} .

Relation (v′) can be simply checked if f is a finite trigonometrical polynomial in the discrete field case, or if f = φ̃, φ ∈ S^{c}, in the generalized field case. (In the case f = φ̃, φ ∈ S^{c}, the following argument works. Put f(x) = φ̃_{1}(x) + iφ̃_{2}(x) with φ_{1},φ_{2} ∈ S. Then I(f) = X(φ_{1}) + iX(φ_{2}), and f_{−} (x) = ¯ φ̃_{1}(−x) − i ¯φ̃_{2}(−x) = φ̃_{1}(x)+i( ˜−φ_{2} (x), hence I(f_{−} ) = X(φ_{1})+iX(−φ_{2}) = X(φ_{1})−iX(φ_{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(χ_{A})\overline{I(χ_{B})} = ∫ χ_{A}(x)\overline{χ_{B}(x)}G(dx) = G(A ∩ B). The remaining properties of Z_{G}(·) are simple consequences of the definition.

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},...,A_{n} are disjoint sets, and property (iii) holds, then

E ( ∑_{j=1}^{n} Z_{G}(A_{j}) − Z_{G}( ⋃_{j=1}^{n} A_{j}) ) \overline{( ∑_{j=1}^{n} Z_{G}(A_{j}) − Z_{G}( ⋃_{j=1}^{n} A_{j}) )} = 0.

Now we introduce the following

Definition of random spectral measure. Let G be a spectral measure. A set of random variables Z_{G}(A), G(A) < ∞, satisfying (i)–(v) is called a (Gaussian) random spectral measure corresponding to the spectral measure G.

Given a Gaussian random spectral measure Z_{G} corresponding to a spectral measure G we define the (one-fold) stochastic integral ∫ f(x)Z_{G}(dx) for an appropriate class of functions f. Let us first consider simple functions of the form f(x) = ∑ c_{i}χ_{A_{i}}(x), where the sum is finite, and G(A_{i}) < ∞ for all indices i. In this case we define

∫ f(x)Z_{G}(dx) = ∑ c_{i}Z_{G}(A_{i}).

Then we have

E |∫ f(x)Z_{G}(dx)|^{2} = ∑ c_{i}c̄_{j}G(A_{i}∩ A_{j}) = ∫ |f(x)|^{2}G(dx).  (3.6)

Since the simple functions are dense in L_{G}^{2}, relation (3.6) enables us to define ∫ f(x)Z_{G}(dx) for all f ∈ L_{G}^{2} via L_{2}-continuity. It can be checked that the expressions

X_{n} = ∫ e^{i(n,x)}Z_{G}(dx), n ∈ ℤ_{ν},  (3.7)

and

X(φ) = ∫ φ̃(x)Z_{G}(dx), φ ∈ S ,  (3.8)

defined with the help of the above defined (random) integral and spectral measure Z_{G} are Gaussian stationary random discrete and generalized field with spectral measure G.

We also have

∫ f(x)Z_{G}(dx) = I(f) for all f ∈ L_{G}^{2}

if we consider the previously defined mapping I(f) with the stationary random fields defined in (3.7) and (3.8). Now we formulate the following

Theorem 3.1. 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 (3.7) or (3.8) holds in the discrete or generalized field case respectively.

Furthermore

B(Z_{G}(A), G(A) < ∞) = \{ \begin{matrix}B(X_{n}, n ∈ ℤ_{ν}) in the discrete field case, \\ B(X(φ), φ ∈ S) in the generalized field case. \end{matrix}  (3.9)

If a stationary Gaussian random field X_{n}, n ∈_{ν}, or X(φ), φ ∈ S, and a random spectral measure Z_{G} satisfy relation (3.7) or (3.8), then we say that this random spectral measure is adapted to this Gaussian random field.

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 (3.7) or (3.8) define the original stationary random field. Since all random variables Z_{G}(A) are measurable with respect to the original random field, relation (3.7) or (3.8) implies (3.9).

To prove the uniqueness, it is enough to observe that because of the linearity and L_{2} continuity of stochastic integrals relation (3.7) or (3.8) implies that

Z_{G}(A) = ∫ χ_{A}(x)Z_{G}(dx) = I(χ_{A})

for a Gaussian random spectral measure corresponding to the spectral measure G appearing in Theorem 3.1. ⊓ ⊔

Finally we list some additional properties of Gaussian random spectral measures.

(vi) The random variables ReZ_{G}(A) are independent of the random variables

ImZ_{G}(A). (vii) Random variables of the form Z_{G}(A ∪ (−A)) are real valued. If the sets

A_{1} ∪ (−A_{1}),…, A_{n}∪ (−A_{n}) are disjoint, then the random variables Z_{G}(A_{1}),…, Z_{G}(A_{n}) are independent.

(viii) If A ∩ (−A) = ∅, then ReZ_{G}(−A) = ReZ_{G}(A), ImZ_{G}(−A) = −ImZ_{G}(A), and the (Gaussian) random variables ReZ_{G}(A) and ImZ_{G}(A) are independent with expectation zero and variance G(A)/2.

These properties easily follow from (i)–(v). Since Z_{G}(·) 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).

EReZ_{G}(A)ImZ_{G}(B)= \frac{1}{4i}E(Z_{G}(A) + \overline{Z_{G}(A)})(Z_{G}(B) − \overline{Z_{G}(B)})

= \frac{1}{4i}E(Z_{G}(A) + Z_{G}(−A))(\overline{Z_{G}(−B)} − \overline{Z_{G}(B)})

= \frac{1}{4i}G(A ∩ (−B)) − \frac{1}{4i}G(A ∩ B)

+\frac{1}{4i}G((−A) ∩ (−B)) − \frac{1}{4i}G((−A) ∩ B) = 0

for all pairs of sets A and B such that G(A) < ∞, G(B) < ∞, since G(D) = G(−D) for all D ∈ B^{ν}. The fact that Z_{G}(A ∪ (−A)) is real valued random variable, and the relations ReZ_{G}(−A) = ReZ_{G}(A), ImZ_{G}(−A) = −ImZ_{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 ∈ B^{ν}.

3.1 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^{ν}/2πℤ_{ν} one can define a (Gaussian) discrete stationary field with correlation function satisfying (3.1) with this measure G. For an even measure G on R^{ν} satisfying (3.3) there exists a (Gaussian) generalized stationary field with correlation function defined in formula (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

Proposition 3C. Let G be a finite measure on the torus R^{ν}/2πℤ_{ν} such that G(A) = G(−A) for all measurable sets A. Then there exists a Gaussian discrete stationary field X_{n}, n ∈ ℤ_{ν}, with expectation zero such that its correlation function r(n) = EX_{k}X_{k+n}, n,k ∈ ℤ_{ν}, is given by formula (3.1) with this measure G.

Let G be a measure on R^{ν} satisfying (3.3) and such that G(A) = G(−A) for all measurable sets A. Then there exists a Gaussian stationary generalized field X(φ), φ ∈ S, with expectation EX(φ) = 0 for all φ ∈ S such that its covariance function EX(φ)X(ψ), φ,ψ ∈ S, satisfies formula (3.2) with this measure G.

Moreover, the correlation function r(n) or EX(φ)X(ψ), φ,ψ ∈ S, determines the measure G uniquely.

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},...,n_{p} ∈ ℤ_{ν} there exists a Gaussian random vector (X_{n_{1}},...,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. ∑_{j,k}c_{j}c_{k}r(n_{j} − n_{k}) ≥ 0

for all real vectors (c_{1},...,c_{p}). This relation holds, because ∑_{j,k}c_{j}c_{k}r(n_{j} − n_{k}) =

∫ |∑_{j} c_{j}e^{i(n_{j},x)}|^{2}G(dx) ≥ 0 by formula (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 (3.2). (Let us observe that the relation G(A) = G(−A) implies that EX(φ)X(ψ) is a real number for all φ, ψ ∈ S, since EX(φ)X(ψ) = \overline{EX(φ)X(ψ)} in this case. In the proof of this identity we exploit that ¯ f̃(x) = f̃(−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 φ_{n} → φ in the topology of the space S, then E[X(φ_{n}) − X(φ)]^{2} = ∫ |φ̃_{n}(x) − φ̃(x)|^{2}G(dx) → 0 as n → ∞, hence property (b) holds. (Here we exploit that the transformation φ → φ̃ is bicontinuous in the space S.) Property (a) also holds, because, as it is not difficult to check with the help of formula (3.2),

E[a_{1}X(φ_{1}) + a_{2}X(φ_{2}) − X(φ(a_{1}φ_{1}+ a_{2}φ_{2})]^{2}

= ∫ |a_{1}φ̃_{1}(x) + a_{2}φ̃_{2}(x) − ({ \atop a_{1}φ_{1}+ a_{2}φ_{2}} )(x)| ^{2}G(dx) = 0.

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 ∫ f(x)G(dx) for all continuous functions f with a bounded support, hence it also determines the measure G. ⊓ ⊔

The Bochner and Bochner–Schwartz theorems enable us to show that the correlation function of all stationary (Gaussian) fields (discrete or generalized one) can be presented 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.

Definition of positive definite functions. Let f(x) be a (complex valued) function on ℤ_{ν} (or on R^{ν}). We say that f(·) is a positive definite function if for all parameters p, complex numbers c_{1},...,c_{p} and points x_{1},...,x_{p} in ℤ_{ν} (or in R^{ν}) the inequality

∑_{j=1}^{p} ∑_{k=1}^{p} c_{j} c̄_{k}f(x_{j} − x_{k}) ≥ 0

holds.

A simple example for positive definite functions is the function f(x) = e^{i(t,x)}, where t ∈ ℤ_{ν} in the discrete, and t ∈ R^{ν} in the continuous case. Bochner’s theorem provides a complete description of positive definite functions.

Bochner’s theorem. (Its original form.) A complex valued function f(x) defined on ℤ_{ν} is positive definite if and only if it can be written in the form f(x) = ∫ e^{i(t,x)}G(dx) for all x ∈ ℤ_{ν} with a finite measure G on the torus R^{ν}/2πℤ_{ν}. The measure G is uniquely determined.

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

It is not difficult to see that the covariance function r(n) = EX_{k}X_{k+n}, (EX_{n} = 0), k,n ∈ ℤ_{ν}, of a stationary (Gaussian) random field X_{n} is a positive definite func-

tion, since ∑_{j,k}c_{j} c̄_{k}r(n_{j} − n_{k}) = E|∑_{j} c_{j}X_{n_{j}}|^{2} > 0 for any vector (c_{1},...,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.

The Bochner–Schwartz theorem yields an analogous representation of positive definite generalized functions in S′as the Fourier transforms of positive generalized functions in S′. It also states a similar result about generalized functions in the space 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 S′and D′, and then we characterize them in a Theorem.

Definition of positive generalized functions. A linear functional F ∈ S′(or F ∈ D′) is called a positive generalized function if for all such φ ∈ S (or φ ∈ D) test functions for which φ(x) ≥ 0 for all x ∈ R^{ν} (F,φ) ≥ 0.

Theorem about the representation of positive generalized functions. All positive generalized functions F ∈ S′can be given in the form (F,φ) = ∫ φ(x)μ(dx), where μ is a polynomially increasing measure on R^{ν}, i.e. it satisfies the relation ∫(1 + |x|^{2})^{−p} μ(dx) < ∞ with some p > 0. Similarly, all positive generalized functions in D′ can be given in the form (F,φ) = ∫ φ(x)μ(dx) with such a measure μ on R^{ν} which is finite in all bounded regions. The positive generalized function F uniquely determines the measure μ in both cases.

We also introduce a rather technical notion and formulate a result about it. Let us remark that if φ ∈ S^{c} and ψ ∈ S^{c}, then also their product φψ ∈ S^{c}. The analogous result also holds in the space D.

Definition of multiplicatively positive generalized functions. A generalized function F ∈ S′(or F ∈ D′) is multiplicatively positive if (F,φφ̄) = (F,|φ|^{2}) ≥ 0 for all φ ∈ S^{c} (or in φ ∈ D).

Theorem about the characterization of multiplicatively positive generalized functions. A generalized function F ∈ S′(or F ∈ D′) is multiplicatively positive if and only if it is positive.

Now I introduce the definition of positive definite generalized functions.

Definition of positive definite generalized functions. A generalized function F ∈ S′ (or F ∈ D′) is positive definite if (F,φ ∗ φ^{∗}) ≥ 0 for all φ ∈ S^{c} (of φ ∈ D), where φ^{∗}(x) = \overline{φ(−x)} , and ∗ denotes convolution, i.e. φ ∗φ^{∗}(x) = ∫ φ(t)\overline{φ(t − x)} dt.

We refer to [15] for an explanation why this definition of positive definite generalized functions is natural. Let us remark that if φ,ψ ∈ S^{c}, then φ ∗ ψ ∈ S^{c}, and the analogous result holds in D. The original version of the Bochner–Schwartz theorem has the following form.

Bochner–Schwartz theorem. (Its original form.) Let F be a positive definite generalized function in the space S′ (or D′). Then it is the Fourier transform of a polynomially increasing measure μ on R^{ν}, i.e. the identity (F,φ) = ∫ φ̃(x)μ(dx) holds for all φ ∈ S^{c} (or φ ∈ D) with a measure μ that satisfies the relation ∫(1 + |x|^{2})^{−p} μ(dx) < ∞ with an appropriate p > 0. The generalized function F uniquely determines the measure μ. On the other hand, if μ is a polynomially increasing measure on R^{ν}, then the formula (F,φ) = ∫ φ̃(x)μ(dx) with φ ∈ S^{c} (or φ ∈ D) defines a positive definite generalized function F in the space S′(or D′).

Remark. It is a remarkable and surprising fact that the class of positive definite generalized functions are represented by the same class of measures μ in the spaces S′ and D′. (In the representation of positive generalized functions the class of measures μ considered in the case of D′is much larger, than in the case of S′.) Let us remark that in the representation of the positive definite generalized functions in D′ the function φ̃ we integrate is not in the class D, but in the space Z consisting of the Fourier transforms of the functions in D.

It is relatively simple to prove the representation of positive definite generalized functions given in the Bochner–Schwartz theorem for the class 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 φ ∗ ψ(x) equals φ̃(t)ψ̃(t), and the Fourier transform of φ^{∗}(x) = \overline{φ(−x)} equals \overline{φ̃(t)}, the Fourier transform of φ ∗ φ^{∗}(x) equals φ̃(t) ¯φ̃(t). Hence the positive definitiveness property of the generalized function F and the definition of the Fourier transform of generalized functions imply that (F̃,φ̃ ¯φ̃) = (2π)^{ν}(F,φ ∗ φ^{∗}) ≥ 0 for all φ ∈ S^{c}. Since every function of S^{c} is the Fourier transform φ̃ of some function φ ∈ S^{c} this implies that F̃ is a multiplicatively positive and as a consequence a positive generalized function in S′. Such generalized functions have a good representation with the help of a polynomially increasing positive measure μ. Since (F,φ) = (2π)^{−ν} (F̃,φ̃) it is not difficult to prove the Bochner–Schwartz theorem for the space S′with the help of this fact. The proof is much harder if the space 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 fields. We still need another important result of Laurent Schwartz which gives useful information about the behaviour of (Hermitian) bilinear functionals in 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.

Definition of Hermitian bilinear and translation invariant Hermitian bilinear functionals in the space S^{c}. A function B(φ,ψ), φ,ψ ∈ S^{c}, is a Hermitian bilinear functional in the space S^{c} if for all fixed ψ ∈ S^{c} B(φ,ψ) is a continuous linear functional of the variable ψ in the topology of S^{c}, and for all fixed φ ∈ S^{c} \overline{B(φ,ψ)} is a continuous linear functional of the variable ψ in the topology of S^{c}.

A Hermitian bilinear functional B(φ,ψ) in S^{c} is translation invariant if it does not change by a simultaneous shift of its variables φ and ψ, i.e. if B(φ(x),ψ(x)) = B(φ(x + h),ψ(x + h)) for all h ∈ R^{ν}.

Definition of positive definite Hermitian bilinear functionals. We say that a Hermitian bilinear functional B(φ,ψ) in S^{c} is positive definite if B(φ,φ) ≥ 0 for all φ ∈ S^{c}.

The next result characterizes the Hermitian bilinear and translation invariant Hermitian bilinear functionals in S^{c}.

Theorem 3D. All Hermitian bilinear functionals B(φ,ψ) in S^{c} can be given in the form B(φ,ψ) = (F_{1},φ(x)\overline{ψ(y)}), φ,ψ ∈ S^{c}, where F_{1} is a continuous linear functional on S^{c}× S^{c}, i.e. it is a generalized function in S_{2ν} ′.

A translation invariant Hermitian bilinear functional in S^{c} can be given in the form B(φ,ψ) = (F,φ ∗ ψ^{∗}), φ,ψ ∈ S^{c}, where F ∈ S, ψ^{∗}(x) = \overline{ψ} (−x), and ∗ denotes convolution.

The Hermitian bilinear form B(φ,ψ) 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} ∈ S^{2ν} ′and F ∈ S′the above formulas define a Hermitian bilinear functional and a translation invariant Hermitian bilinear functional in S^{c} respectively.

Let us consider a Gaussian generalized random field X(φ), φ ∈ S, with expectation zero together with its correlation function B(φ,ψ) = EXφ)X(ψ), φ,ψ ∈ S. More precisely, let us consider the complexification X(φ_{1}+ iφ_{2}) = X(φ_{1}) + iX(φ_{2}) of this random field and its correlation function B(φ,ψ) = EX(φ)\overline{X(ψ)}, φ,ψ ∈ S^{c}. This correlation function B(φ,ψ) is a translation invariant Hermitian bilinear functional in S^{c}, hence it can be written in the form B(φ,ψ) = (F,φ ∗ ψ^{∗}) with an appropriate F ∈ S′. Moreover, B(φ,φ) ≥ 0 for all φ ∈ S^{c}, and this means that the generalized function F ∈ S′corresponding to B(φ,ψ) is positive definite. Hence the Bochner–Schwartz theorem can be applied for it, and it yields that

EX(φ)X(ψ) = ∫ { \atop φ ∗ ψ^{∗}}(x)G(dx) = ∫ φ̃(x) ¯ψ̃(x)G(dx) for all φ, ψ ∈ S^{c}

with a uniquely determined, polynomially increasing measure G on R^{ν}. Now we prove with the help of these results Theorem 3B.

Proof of Theorem 3B. We have already proved relations (3.2) and (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 φ ∈ S the random variable X(φ) is also real valued. Hence if φ,ψ ∈ S, then EX(φ)X(ψ) = \overline{EX(φ)X(ψ)}. Besides, φ̃(−x) = ¯φ̃(x) and ψ̃(−x) = ¯ ψ̃(x) in this case. Hence

∫ φ̃(x) ¯ψ̃(x)G(dx)= ∫ ¯ φ̃(x)ψ̃(x)G(dx)
= ∫ φ̃(−x) ¯ψ̃(−x)G(dx) = ∫ φ̃(x) ¯ψ̃(x)G^{−} (dx)

for all φ,ψ ∈ S, where G^{−} (A) = G(−A) for all A ∈ B^{ν}. This relation implies that the measures G and G^{−} agree. The proof of Theorem 3B is completed. ⊓ ⊔

Multiple Wiener–Itô integrals

In this chapter we define the so-called multiple Wiener–Itô integrals, and we prove their most important properties with the help of Itô’s formula, whose proof is postponed to the next chapter. More precisely, we discuss in this chapter a modified version of the Wiener–Itô 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–Itô integral seems to be a more useful tool than the original one or the Wick polynomials, 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 real Hilbert spaces H̄_{G}^{n} and H_{G}^{n}, n = 1,2,.... We have f_{n} ∈ H̄_{G}^{n} if and only if f_{n} = f_{n}(x_{1},...,x_{n}), x_{j} ∈ R^{ν}, j = 1,2,...,n, is a complex valued function of n variables, and

(a) f_{n}(−x_{1},...,−x_{n}) = \overline{f_{n}(x_{1},...,x_{n})},

(b) ∥f_{n}∥^{2} = ∫ |f_{n}(x_{1},...,x_{n})|^{2}G(dx_{1})...G(dx_{n}) < ∞.

Relation (b) also defines the norm in H̄_{G}^{n}. The subspace H_{G}^{n} ⊂ H̄_{G}^{n} contains those functions f_{n}∈ H̄_{G}^{n} which are invariant under permutations of their arguments, i.e.

(c) f_{n}(x_{π(1)},...,x_{π(n)})) = f_{n}(x_{1},...,x_{n}) for all π ∈ Π_{n}, where Π_{n} denotes the group of all permutations of the set {1,2,...,n}.

The norm in H_{G}^{n} is defined in the same way as in H̄_{G}^{n}. Moreover, the scalar product is also similarly defined, namely if f, g ∈ H̄_{G}^{n}, then

(f,g) = ∫ f(x_{1},...,x_{n})\overline{g(x_{1},...,x_{n})}G(dx_{1})...G(dx_{n})
= ∫ f(x_{1},...,x_{n})g(−x_{1},...,−x_{n})G(dx_{1})...G(dx_{n}).

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 ∈ H̄_{G}^{n}. This means that H̄_{G}^{n} is a real Hilbert space. We also define H_{G}^{0} = H̄_{G}^{0} as the space of real constants with the norm ∥c∥ = |c|. We remark that H̄_{G}^{n} is actually the n-fold direct product of H_{G}^{1}, while H_{G}^{n} is the n-fold symmetrical direct product of H_{G}^{1}. Condition (a) means heuristically that f_{n} is the Fourier transform of a real valued function.

Finally we define the so-called Fock space ExpH_{G} whose elements are sequences of functions f = (f_{0},f_{1},...), f_{n} ∈ H_{G}^{n} for all n = 0,1,2,..., such that

∥f∥^{2} = ∑_{n=0}^{∞} \frac{1}{n!}∥f_{n}∥^{2} < ∞.

Given a function f ∈ H̄_{G}^{n} we define Symf as

Symf(x_{1},...,x_{n}) = \frac{1}{n!} ∑_{π∈Π_{n}} f(x_{π(1)},...,x_{π(n)}).

Clearly, Symf ∈ H_{G}^{n}, and

∥Symf∥ ≤ ∥f∥.  (4.1)

Let Z_{G} be a Gaussian random spectral measure corresponding to the spectral measure G on a probability space (Ω,A ,P). We shall define the n-fold Wiener–Itô integrals

I_{G}(f_{n}) = \frac{1}{n!} ∫ f_{n}(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n}) for f_{n} ∈ H̄_{G}^{n}

and

I_{G}(f) = ∑_{n=0}^{∞} I_{G}(f_{n}) for f = (f_{0},f_{1},...) ∈ ExpH_{G}.

We shall see that I_{G}(f_{n}) = I_{G}(Symf_{n}) for all f_{n} ∈ H̄_{G}^{n}. Therefore, it would have been sufficient to define the Wiener–Itô integral only for functions in H_{G}^{n}. Nevertheless, some arguments become simpler if we work in H̄_{G}^{n}. In the definition of Wiener–Itô integrals first we restrict ourselves to the case when the spectral measure is non-atomic, i.e. G({x}) = 0 for all x ∈ R^{ν}. 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^{ν}, define a subclass ˆ ¯H_{G}^{n} ⊂ H̄_{G}^{n} of simple functions with their help, and define the Wiener–Itô integrals for the functions of this subclass.

Definition of regular systems and the class of simple functions. Let

D = {Δ_{j}, j = ±1,±2,...,±N}

be a finite collection of bounded, measurable sets in R^{ν} indexed by the integers ±1,...,±N. We say that D is a regular system if Δ_{j} = −Δ_{−j} , and Δ_{j} ∩ Δ_{l} = ∅ if j ≠ l for all j,l = ±1,±2,...,±N. A function f ∈ H̄_{G}^{n} is adapted to this system D if f(x_{1},...,x_{n}) is constant on the sets Δ_{j_{1}} × Δ_{j_{2}} × ···× Δ_{j_{n}}, j_{l} = ±1,...,±N, l = 1,2,...,n, it vanishes outside these sets and also on the sets for which j_{l} = ±j_{l′} for some l ≠ l′.

A function f ∈ H̄_{G}^{n} is in the class ˆ ¯H_{G}^{n} of simple functions, and a (symmetric) function f ∈ H_{G}^{n} is in the class Ĥ_{G}^{n} of simple symmetric functions if it is adapted to some regular system D = {Δ_{j}, j = ±1,...,±N}.

Definition of Wiener–Itô integral of simple functions. Let a simple function f ∈ ˆ¯H_{G}^{n} be adapted to some regular systems D = {Δ_{j}, j ± 1,...,±N}. Its Wiener–Itô integral with respect to the random spectral measure Z_{G} is defined as

∫ f(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n})  (4.2)

= n!I_{G}(f) = ∑_{\begin{matrix}j_{l}=±1,...,±N \\ l=1,2,...,n \end{matrix}} f(x_{j_{1}},...,x_{j_{n}})Z_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}}),

where x_{j_{l}} ∈ Δ_{j_{l}}, j_{l} = ±1,...,±N, l = 1,...,n.

We remark that although the regular system D to which f is adapted, is not uniquely determined (the elements of D can be divided to smaller sets), the integral defined in (4.2) is meaningful, i.e. it does not depend on the choice of D. This can be seen by observing that a refinement of a regular system D to which the function f is adapted yields the same value for the sum defining n!I_{G}(f) in formula (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}},...,x_{j_{n}})Z_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}}) in the sum at the right-hand side of formula (4.2) corresponding to the original regular system equals the sum of all such terms f(x_{j_{1}},...,x_{j_{n}})Z_{G}(Δ′_{j′_{1}})··· Z_{G}(Δ′_{j′_{n}}) in the sum corresponding to the refined partition for which Δ′_{j′_{1}} × ··· × Δ′_{j′_{n}} ⊂ Δ_{j_{1}} × ··· × Δ_{j_{n}}.

By property (vii) of the random spectral measures all products

Z_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})

with non-zero coefficient in (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 Δ_{j_{1}} × ···× Δ_{j_{n}} if j_{l} = ±j_{l′} for some l ≠ 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,...,n, l ≠ l′, from the domain of integration. (Let us observe that in this case, — unlike to the definition of the original Wiener–Itô integrals discussed in Chapter 7, — we omitted also the hyperplanes x_{l} = −x_{l′} and not only the hyperplanes x_{l} = x_{l}. l ≠ l′, from the domain of integration.) Property (a) of the functions in H̄_{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 ∈ ˆ ¯H_{G}^{n}. The relation

EI_{G}(f) = 0, for f ∈ ˆ ¯H_{G}^{n}, n = 1,2,...  (4.3)

also holds. Let Ĥ_{G}^{n} = H_{G}^{n}∩ ˆ ¯H_{G}^{n}. If f ∈ ˆ ¯H_{G}^{n}, then Symf ∈ Ĥ_{G}^{n}, and

I_{G}(f) = I_{G}(Symf).  (4.4)

Relation (4.4) follows immediately from the observation that Z_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}}) = Z_{G}(Δ_{π(j_{1})})··· Z_{G}(Δ_{π(j_{n})}) for all π ∈ Π_{n}. We also claim that

EI_{G}(f)^{2} ≤ \frac{1}{n!}∥f∥^{2} for f ∈ ˆ ¯H_{G}^{n},  (4.5)

and

EI_{G}(f)^{2} = \frac{1}{n!}∥f∥^{2} for f ∈ Ĥ_{G}^{n} .  (4.6)

Because of (4.1) and (4.4) it is enough to check (4.6).

Let D be a regular system of sets in R^{ν}, j_{1},...,j_{n} and k_{1},...,k_{n} be indices such that j_{l} ≠ ±j_{l′}, k_{l} ≠ ±k_{l′} if l ≠ l′. Then

EZ_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})\overline{Z_{G}(Δ_{k_{1}})··· Z_{G}(Δ_{k_{n}})}

= \{ \begin{matrix}G(Δ_{j_{1}})··· G(Δ_{j_{n}}) if \{j_{1},...,j_{n}\} = \{k_{1},...,k_{n}\}, \\ 0 otherwise. \end{matrix}

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},...,j_{n}} ≠ {k_{1},...,k_{n}}, then there is an index l such that either j_{l} ≠ ±k_{l′} for all 1 ≤ l′≤ n, or there exists an index l′, 1 ≤ l′≤ n, such that j_{l} = −k_{l′}. In the first case Z_{G}(Δ_{j_{l}}) is independent of the remaining coordinates of the vector (Z_{G}(Δ_{j_{1}}),...,Z_{G}(Δ_{j_{n}}),\overline{Z_{G}(Δ_{k_{1}})},...,\overline{Z_{G}(Δ_{k_{n}})}), and EZ_{G}(Δ_{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}(Δ_{j_{l}}) and Z_{G}(−Δ_{j_{l}}) are not independent. In this case we can state that since j_{p} ≠ ±j_{l} if p ≠ l, and k_{q} ≠ ±j_{l} if q ≠ l′, the vector (Z_{G}(Δ_{j_{l}}),Z_{G}(−Δ_{j_{l}})) is independent of the remaining coordinates of the above random vector. On the other hand, the product Z_{G}(Δ_{j_{l}})\overline{Z_{G}(−Δ_{j_{l}}} ) has zero expectation, since EZ_{G}(Δ_{j_{l}})\overline{Z_{G}(−Δ_{j_{l}})} = G(Δ_{j_{l}} ∩ (−Δ_{j_{l}})) = 0 by property (iii) of the random spectral measures and the relation Δ_{j_{l}} ∩ (−Δ_{j_{l}}) = ∅. Hence the expectation of the considered product equals zero also in this case. If {j_{1},...,j_{n}} = {k_{1},...,k_{n}}, then

EZ_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})\overline{Z_{G}(Δ_{k_{1}})··· Z_{G}(Δ_{k_{n}})} = ∏_{l=1}^{n} EZ_{G}(Δ_{j_{l}})\overline{Z_{G}(Δ_{j_{l}})} = ∏_{l=1}^{n} G(Δ_{j_{l}}).

Therefore for a function f ∈ Ĥ_{G}^{n}

EI_{G}(f)^{2} = ( \frac{1}{n}) ^{2}∑ ∑ f(x_{j_{1}},...,x_{j_{n}})\overline{f(x_{k_{1}},...,x_{k_{n}})}
EZ_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})\overline{Z_{G}(Δ_{k_{1}})··· Z_{G}(Δ_{k_{n}})}

= ( \frac{1}{n!}) ^{2}∑ |f(x_{j_{1}},...,x_{j_{n}})|^{2}G(Δ_{j_{1}})··· G(Δ_{j_{n}}) · n!

= \frac{1}{n!} ∫ |f(x_{1},...,x_{n})|^{2}G(dx_{1})··· G(dx_{n}) = \frac{1}{n!}∥f∥^{2}.

We claim that Wiener–Itô integrals of different order are uncorrelated. More explicitly, take two functions f ∈ ˆ ¯H_{G}^{n} and f′∈ ˆ ¯H_{G}^{n′}such that n ≠ n′. Then we have

EI_{G}(f)I_{G}(f′) = 0 if f ∈ ˆ ¯H_{G}^{n}, f′∈ ˆ ¯H_{G}^{n′}, and n ≠ n′.  (4.7)

To see this relation observe that a regular system 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}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})\overline{Z_{G}(Δ_{k_{1}})··· Z_{G}(Δ_{k_{n′}})} = 0

for all sets of indices {j_{1},...,j_{n}} and {k_{1},...,k_{n′}} if n ≠ n′, hence the sum expressing EI_{G}(f)I_{G}(f′) in this case equals zero.

We extend the definition of Wiener–Itô 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 which make its reading unpleasant.

Lemma 4.1. The class of simple functions ˆ ¯H_{G}^{n} is dense in the (real) Hilbert space H̄_{G}^{n}, and the class of symmetric simple function Ĥ_{G}^{n} is dense in the (real) Hilbert space H_{G}^{n}.

Proof of Lemma 4.1. It is enough to show that ˆ ¯H_{G}^{n} is dense in the Hilbert space H̄_{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 Statement A and then to a Statement B. Finally we prove Statement B. In Statement A we claim that the indicator function χ_{A} of a bounded set A ∈ B^{nν} such that A = −A can be well approximated by a function of the form g = χ_{B} ∈ ˆ ¯H_{G}^{n}, where χ_{B} is the indicator function of an appropriate set B. Actually we formulate Statement A 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 ∈ H̄_{G}^{n} by a function in g ∈ ˆ ¯H_{G}^{n} to Statement A.

Statement A. Let A ∈ B^{nν} be a bounded, symmetric set, i.e. let A = −A. Then for any ε > 0 there is a function g ∈ ˆ ¯H_{G}^{n} such that g = χ_{B} with some set B ∈ B^{nν}, i.e. g is the indicator function of a set B, such that the inequality ∥g − χ_{A}∥ < ε holds with the norm of the space H̄_{G}^{n}. (Here χ_{A}denotes the indicator function of the set A, and we have χ_{A}∈ H̄_{G}^{n}.)

If the set A can be written in the form A = A_{1}∪(−A_{1}) with such a set A_{1} for which the sets A_{1} and −A_{1} have a positive distance from each other, i.e. ρ(A_{1},−A_{1}) = inf_{x∈A_{1},y∈−A_{1}} ρ(x,y) > δ, with some δ > 0, where ρ denotes the Euclidean distance

in R^{nν}, then a good approximation of χ_{A} can be given with such a function g = χ_{B∪(−B)} ∈ ˆ ¯H_{G}^{n} for which the sets B and −B are separated from each other. More

explicitly, for all ε > 0 there is a set B ∈ B^{nν} such that B ⊂ A_{1}^{δ/2} = {x: ρ(x,A_{1}) ≤ \frac{δ}{2}\}, g = χ_{B∪(−B)} ∈ ˆ ¯H_{G}^{n}, and G^{n}(A_{1}Δ B) < \frac{ε}{2}. Here AΔB 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^{nν}. (The above properties of the set B imply that the function g = χ_{B∪(−B)} ∈ ˆ ¯H_{G}^{n} satisfies the relation ∥g − χ_{A}∥ < ε.)

To justify the reduction of Lemma 4.1 to Statement A let us observe that if two functions f_{1} ∈ H̄_{G}^{n} and f_{2} ∈ H̄_{G}^{n} can be arbitrarily well approximated by functions from ˆ ¯H_{G}^{n} in the norm of this space, then the same relation holds for any linear combination c_{1}f_{1}+ c_{2}f_{2} with real coefficients c_{1} and c_{2}. (If the functions f_{i} are approximated by some functions g_{i} ∈ ˆ ¯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 ∈ H̄_{G}^{n} by functions g ∈ ˆ ¯H_{G}^{n} can be reduced to the proof about the arbitrarily good approximability of its real part Ref ∈ H̄_{G}^{n} and its imaginary part Imf ∈ H̄_{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 H̄_{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 ∈ B^{nν} such that A = −A (if f is real valued), or it takes three values, the value i on a set A_{1} ∈ B^{nν}, the value −i on the set −A_{1}, and it equals zero on R^{nν} \ (A_{1}∪ (−A_{1})) (if f is purely imaginary valued). Besides, the inequalities G^{n}(A) < ∞ and G^{n}(A_{1}) < ∞ hold. We may even assume that A and A_{1} are bounded sets, because G^{n}(A) = lim_{K→∞} G^{n}(A∩[−K,K]^{nν}), and the same argument applies for A_{1}.

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 ±i and zero, observe that the sets A_{1} = {x: f(x) = i} and −A_{1} = {x: 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} ⊂ A_{1}, N = 1,2,..., for which lim_{N→∞} G^{n}(A \ (K_{N}∪ (−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: f(x) = i} = K_{N} and −A_{1} = {x: f(x) = −i} = −K_{N} with one of the above defined sets K_{N} with a sufficiently large index N. To get a good approximation in this case, take A_{1} = K_{N} and apply the second part of Statement A for the indicator function χ_{A}= χ_{K_{N}∪(−K_{N})} with the choice A_{1} = K_{N}. We get that there exists a function

g = χ_{B∪(−B)} ∈ ˆ ¯H_{G}^{n} such that B ⊂ A_{1}^{δ/2} with a number δ > 0 for which the relation ρ(K_{N},−K_{N}) > δ holds, and G^{n}(A_{1}Δ B) < \frac{ε}{2}. Then we define with the help of the above set B the function ḡ∈ ˆ ¯H_{G}^{n} as ḡ(x) = i if x ∈ B, ḡ(x) = −i if x ∈ −B and ḡ(x) = 0 otherwise. The definition of the function ḡ(·) is meaningful, since B ∩ (−B) = ∅, and it yields a sufficiently good approximation of the function f(·).

In the next step we reduce the proof of Statement A to the proof of a result called Statement B. We show that to prove Statement A it is enough to prove the good approximability of some very special (and relatively simple) indicator functions χ_{B} ∈ H̄_{G}^{n} by a function g ∈ ˆ ¯H_{G}^{n}.

Statement B. Let B = D_{1}× ··· × D_{n} be the direct product of bounded sets D_{j} ∈ B^{ν} such that D_{j} ∩ (−D_{j}) = ∅ for all 1 ≤ j ≤ n. Then for all ε > 0 there is a set F ⊂ B ∪ (−B), F ∈ B^{nν} such that χ_{F} ∈ ˆ ¯H_{G}^{n}, and ∥χ_{B∪(−B)} − χ_{F}∥ ≤ ε, with the norm of the space H̄_{G}^{n}.

To deduce Statement A from Statement B let us first remark that we may reduce our attention to such sets A in 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 η > 0 with the property A ∩ K(η) = ∅, where K(η) = ⋃_{j=1}^{n} K_{j}(η) with

K_{j}(η) = {(x_{1},...,x_{n}): x_{l} ∈ R^{ν}, l = 1,...,n, ρ(x_{j},0) ≤ η}. To see our right to make such a reduction observe that the relation G({0}) = 0 implies that lim_{η→0}G^{n}(K(η)) =

0, hence lim_{η→0}G^{n}(A\K(η)) = 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.

First we formulate a result which we prove somewhat later, and show that the proof of Statement A can be reduced to that of Statement B with its help. We claim that for all numbers ε > 0, δ̄ > 0 and bounded sets A ∈ B^{nν} such that A = −A, and A ∪ K(η) = ∅ there is a finite sequence of bounded sets B_{j} ∈ B^{nν}, j = ±1,...,±N, with the following properties. The sets B_{j} are disjoint, B_{−j} = −B_{j}, j = ±1,...,±N, each set B_{j} can be written in the form B_{j} = D_{1}^{(j)} × ··· × D_{n}^{(j)} with D_{k}^{(j)} ∈ B^{ν}, and D_{k}^{(−j)} ∩ (−D_{k}^{(j)}) = ∅ for all 1 ≤ j ≤ N and 1 ≤ k ≤ n, the diameter d(B_{j}) = sup{ρ(x,y): x,y ∈ B_{j}} of the sets B_{j} has the bound d(B_{j}) ≤ δ̄ for all

1 ≤ j ≤ N, and finally the set B = ⋃_{j=1}^{N} (B_{j} ∪B_{−j} ) satisfies the relation G^{n}(AΔB) ≤ ε.

Indeed, since we can choose ε > 0 arbitrarily small, the above result together with the application of Statement B for all functions χ_{B_{j},∪(−B_{j})} , 1 ≤ j ≤ N, supplies an

arbitrarily good approximation of the function χ_{A}by a function of the form ∑_{j=1}^{N} χ_{F_{j}} ∈

ˆ¯H_{G}^{n} in the norm of the space H̄_{G}^{n}. Moreover, the random variable ∑_{j=1}^{N} χ_{F_{j}} ∈ ˆ ¯H_{G}^{n} agrees with the indicator function of the set ⋃_{j=1}^{N} F_{j}, since the sets B_{j}, j = ±1,...,±N,

are disjoint, and F_{j} ⊂ B_{j} ∪ B_{−j} .

If the set A can be written in the form A = A_{1}∪(−A_{1}) such that ρ(A_{1},−A_{1}) > δ, 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}) ≤ δ̄ with some δ̄ < \frac{δ}{2} for all 1 ≤ j ≤ N. We may assume that A ∩ (B_{j} ∪ B_{−j} ) ≠ ∅ for all indices j, since we can omit those sets B_{j} ∪ B_{−j} which do not have this property. Since d(B_{j}) < \frac{δ}{2}, 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} ⊂ A_{1}^{δ/2} and B_{−j} ⊂ (−A_{1})^{δ/2} for all 1 ≤ j ≤ N. Then the set B = ⋃_{j=1}^{N} (B_{j} ∩F_{j})

and the function g = χ_{B∪(−B)} satisfy the second part of Statement A.

To find a sequence B_{j}, j = ±1,...,±N, for a set A such that A = −A, and A ∪ K(η) = ∅ 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_{1}^{(j)} × ···× D_{n}^{(j)}, D_{l}^{(j)} ∈ B^{ν}, whose union B = ⋃ B_{j} satisfies the relation G^{n}(AΔ B) < \frac{ε}{2}. 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} ∩ A ≠ ∅ for all sets B_{j}. Besides, we may assume, by dividing the sets D_{l}^{(j)} appearing in the definition of the sets B_{j} into smaller sets if this is needed that their diameter d(D_{l}^{(j)}) < max(\frac{η}{2}, \frac{δ̄}{n}). This implies because of the relation A ∩ K(η) = ∅ that D_{l}^{(j)} ∩ (−D_{l}^{(j)}) = ∅ for all j and 1 ≤ l ≤ n. The above constructed sets B_{j} may be not 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 the following partition of R^{ν} for all 1 ≤ l ≤ n with the help of the sets D_{l}^{(j)}, 1 ≤ j ≤ N. For a fixed number l this partition consists of all sets D̄_{r}^{(l)} of the form D̄_{r}^{(l)} = ⋂_{1≤j≤N} F_{l,j}^{r(j)}, where the indices r are sequences (r(1),...,r(N))

of length N with r(j) = 1,2 or 3, 1 ≤ j ≤ N, and F_{l,j}^{(1)} = D_{l}^{(j)}, F_{l,j}^{(2)} = −D_{l}^{(j)}, F_{l,j}^{(3)} =

R^{ν} \ (D_{l}^{(j)} ∪ (−D_{l}^{(j)})). Then B can be represented as the union of those sets of the form D̄_{r_{1}}^{(1)}× ··· × D̄_{r_{n}}^{(n)} which are contained in B.

Proof of statement B. To prove this result we show that for all ε̄ > 0 there is a regular system D = {Δ_{l}, l = ±1,...,±N} such that all sets D_{j} and −D_{j}, 1 ≤ j ≤ n, appearing in the formulation of Statement B can be expressed as the union of some elements Δ_{l} of D, and G(Δ_{l}) ≤ ε̄ for all Δ_{l} ∈ D.

First we prove a weakened version of this statement. We show that there is a regular system D̄ = {Δ′_{l}, l = ±1,...,±N′} such that all sets D_{j} and −D_{j} can be expressed as the union of some sets Δ′_{l}of D̄. But we have no control on the measure G(Δ′_{l}) of the elements of this regular system. To get such a regular system we define the sets Δ′(ε_{s}, 1 ≤ |s| ≤ n) = D_{1}^{ε_{1}} ∩ (−D_{1})^{ε_{−1}} ∩ ··· ∩ D_{n}^{ε_{n}}∩ (−D_{n})^{ε_{−n}} for all vectors (ε_{s}, 1 ≤ |s| ≤ n) such that ε_{s} = ±1 for all 1 ≤ |s| ≤ n, and the vector (ε_{s}, 1 ≤ |s| ≤ n) contains at least one coordinate +1, and D^{1} = D, D^{−1} = R^{ν} \ D for all sets D ∈ B^{ν}. Then taking an appropriate reindexation of the sets Δ′(ε_{s}, 1 ≤ |s| ≤ n) we get a regular system D̄ with the desired properties. (In this construction the sets Δ′(ε_{s}, 1 ≤ |s| ≤ n) are disjoint, and during their reindexation we drop those of them which equal the empty set.) To see that D̄ with a good indexation is a regular system observe that for a set Δ′_{l}= Δ′(ε_{s}, 1 ≤ |s| ≤ n) ∈ D̄ we have −Δ′_{l}= Δ′(ε_{−s} , 1 ≤ |s| ≤ n) ∈ D̄, and Δ′_{l}∩ (−Δ′_{l}) ⊂ D_{j} ∩ (−D_{j}) = ∅ with some index 1 ≤ j ≤ n. (We had to exclude the possibility Δ_{l} = −Δ_{l}.)

Next we show that by appropriately refining the above regular system D̄ we can get such a regular system D = {Δ_{l}, l = ±1,...,±N} which satisfies also the property G(Δ_{l}) ≤ ε̄ for all Δ_{l} ∈ D. To show this let us observe that there is a finite

partition {E_{1},...,E_{l}} of ⋃_{j=1}^{n} (D_{j} ∪ (−D_{j})) such that G(E_{j}) ≤ ε̄ for all 1 ≤ j ≤ l. In-

deed, the closure of D = ⋃_{j=1}^{n} (D_{j} ∪(−D_{j})) can be covered by open sets H_{i} ⊂ R^{ν} such

that G(H_{i}) ≤ ε̄ 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},...,E_{l}} of ⋃_{j=1}^{n} (D_{j} ∪(−D_{j})) with the desired

properties.

Then we can make the following construction with the help of the above sets E_{j} and Δ′_{l}. Take a pair of elements (Δ′_{l},Δ′_{−l} ) = (Δ′_{l},−Δ′_{l}), of D̄, and split up the set Δ′_{l} with the help of the sets E_{j} to the union of finitely many disjoint sets of the form Δ_{l,j} = Δ′_{l}∩E_{j}. Then G(Δ_{l,j}) < ε̄ for all sets Δ_{l,j}, and we can write the set Δ′_{−l} as the union of the disjoint sets −Δ_{l,j}. By applying this procedure for all pairs (Δ′_{l},Δ′_{−l} ) and by reindexing the sets Δ_{l,j} obtained by this procedure in an appropriate way we get a regular system D with the desired properties.

Let us write B ∪ (−B) as the union of products of sets of the form Δ_{l_{1}} × ··· × Δ_{l_{n}} with sets Δ_{l_{j}} ∈ D, 1 ≤ j ≤ n, and let us discard those products for which l_{j} = ±l_{j′} for some pair (j, j′), j ≠ j′. We define the set F about which we claim that it satisfies Property B as the union of the remaining sets Δ_{l_{1}} × ··· × Δ_{l_{n}}. Then χ_{F} ∈ ˆ ¯H_{G}^{n}. Hence to prove that Statement B holds with this set F if ε̄> 0 is chosen sufficiently small it is enough to show that the sum of the terms G(Δ_{l_{1}})··· G(Δ_{l_{n}}) for which l_{j} = ±l_{j′} with some j ≠ j′is less than n^{2}ε̄M^{n−1} , where M = maxG(D_{j} ∪(−D_{j})) = 2maxG(D_{j}). To see this observe that for a fixed pair (j,j′), j ≠ j′, the sum of all products G(Δ_{l_{1}})··· G(Δ_{l_{n}}) such that l_{j} = l_{j′} can be bounded by ε̄M^{n−1} , and the same estimate holds if summation is taken for products with the property l_{j} = −l_{j′}. In-

deed, each term of this sum can be bounded by ε̄G^{n−1} ( ∏_{1≤p≤n,p≠j} Δ_{l_{p}}) , and the

events whose G^{n−1} measure is considered in the investigated sum are disjoint. Beside this their union is in the product set ∏_{1≤p≤n,p≠j} (D_{p} ∪ D_{−p} ), whose measure is

bounded by M^{n−1} . Lemma 4.1 is proved ⊓ ⊔

As the transformation I_{G}(f) is a contraction from ˆ ¯H_{G}^{n} into L_{2}(G,A ,P), it can uniquely be extended to the closure of ˆ ¯H_{G}^{n}, i.e. to H̄_{G}^{n}. We define the n-fold Wiener– Itô integral in the general case via this extension. The expression I_{G}(f) is a real valued random variable for all f ∈ H̄_{G}^{n}, and relations (4.3), (4.4), (4.5), (4.6) and (4.7) remain valid for f, f′∈ H̄_{G}^{n} or f ∈ H_{G}^{n} instead of f, f′∈ ˆ ¯H_{G}^{n} of f ∈ Ĥ_{G}^{n}. Relations (4.6) and (4.7) imply that the transformation I_{G}: ExpH_{G} → L^{2}(Ω,A ,P) is an isometry. We shall show that also the following result holds.

Theorem 4.2. 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}: ExpH_{G} → H is unitary. The transformation (n!)^{1/2}I_{G}: H_{G}^{n} → H_{n} is also unitary.

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

Theorem 4.3. (Itô’s formula.) Let φ_{1},...,φ_{m}, φ_{j} ∈ H_{G}^{1}, 1 ≤ j ≤ m, be an orthonormal system in L_{G}^{2}. Let some positive integers j_{1},...,j_{m} be given, and let j_{1} + ···+ j_{m} = N. Define for all i = 1,...,N the function g_{i} as g_{i} = φ_{s} for j_{1}+ ··· + j_{s−1} < i ≤ j_{1}+ ··· + j_{s}, 1 ≤ s ≤ m. (In particular, g_{i} = φ_{1} for 0 < i ≤ j_{1}.) Then

H_{j_{1}} ( ∫ φ_{1}(x)Z_{G}(dx)) ··· H_{j_{m}} ( ∫ φ_{m}(x)Z_{G}(dx))
= ∫ g_{1}(x_{1})··· g_{N}(x_{N})Z_{G}(dx_{1})··· Z_{G}(dx_{N})
= ∫ Sym[g_{1}(x_{1})··· g_{N}(x_{N})]Z_{G}(dx_{1})··· Z_{G}(dx_{N})

(H_{j}(x) denotes again the j-th Hermite polynomial with leading coefficient 1.)

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 ExpH_{G} to H and from H_{G}^{n} to H_{n}.

The one-fold integral I_{G}(f), f ∈ 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}(φ̃) = X(φ), φ ∈ S, in the generalized field case. Hence I_{G}: H_{G}^{1} → H_{1} is a unitary transformation. Let φ_{1},φ_{2},... be a complete orthonormal basis in H_{G}^{1}. Then ξ_{j} = ∫ φ_{j}(x)Z_{G}(dx), j = 1,2,..., is a complete orthonormal basis in H_{G}^{1}. Itô’s formula implies that for all sets of positive integers (j_{1},...,j_{m}) the random variable H_{j_{1}}(ξ_{1})··· H_{j_{m}}(ξ_{m}) can be written as a j_{1} + ··· + j_{m}-fold Wiener–Itô integral. Therefore Theorem 2.1 implies that the image of ExpH_{G} is the whole space H , and I_{G}: ExpH_{G} is unitary.

The image of H_{G}^{n} contains H_{n}, because of Corollary 2.4 and Itó’s formula. Since these images are orthogonal for different n, formula (2.1) implies that the image of H_{G}^{n} coincides with H_{n}. Hence (n!)^{1/2}I_{G}: H_{G}^{n} → H_{n} is a unitary transformation. ⊓ ⊔

The next result describes the action of shift transformations in H . We know by Theorem 4.2 that all η ∈ H can be written in the form

η = f_{0}+ ∑_{n=1}^{∞} \frac{1}{n!} ∫ f_{n}(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n})  (4.8)

with f = (f_{0},f_{1},...) ∈ ExpH_{G} in a unique way, where Z_{G} is the random measure adapted to the stationary Gaussian field.

Theorem 4.4. Let η ∈ H have the form (4.8). Then

T_{t}η = f_{0}+ ∑_{n=1}^{∞} \frac{1}{n!} ∫ e^{i(t,x_{1}+··· +x_{n})} f_{n}(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n})

for all t ∈ R^{ν} in the generalized field and for all t ∈ ℤ_{ν} in the discrete field case.

Proof of Theorem 4.4. Because of formulas (3.7) and (3.8) and the definition of the shift operator T_{t} we have

T_{t}( ∫ e^{i(n,x)}Z_{G}(dx)) = T_{t}X_{n} = X_{n+t} = ∫ e^{i(t,x)}e^{i(n,x)}Z_{G}(dx), t ∈ ℤ_{ν},

and because of the identity T̃_{t}φ(x) = ∫ e^{(i(u,x)}φ(u − t)du = e^{i(t,x)}φ̃(x) for φ ∈ S

T_{t}( ∫ φ̃(x)Z_{G}(dx))= T_{t}X(φ) = X(T_{t}φ)
= ∫ e^{i(t,x)}φ̃(x)Z_{G}(dx), φ ∈ S , t ∈ R^{ν},

in the discrete and generalized field cases respectively. Hence

T_{t}( ∫ f(x)Z_{G}(dx)) = ∫ e^{i(t,x)}f(x)Z_{G}(dx) if f ∈ H_{G}^{1}

for all t ∈ ℤ_{ν} in the discrete field and for all t ∈ R^{ν} in the generalized field case. This means that Theorem 4.4 holds in the special case when η is a one-fold Wiener–Itô integral. Let f_{1}(x),...,f_{m}(x) be an orthogonal system in H_{G}^{1}. The set of functions e^{i(t,x)}f_{1}(x),...,e^{i(t,x)}f_{m}(x) is also an orthogonal system in H_{G}^{1}. (t ∈ ℤ_{ν} in the discrete and t ∈ R^{ν} in the generalized field case.) Hence Itô’s formula implies that Theorem 4.4 also holds for random variables of the form

η = H_{j_{1}} ( ∫ f_{1}(x)Z_{G}(dx)) ··· H_{j_{m}} ( ∫ f_{m}(x)Z_{G}(dx))

and for their finite linear combinations. Since these linear combinations are dense in H Theorem 4.4 holds true. ⊓ ⊔

The next result is a formula for the change of variables in Wiener–Itô integrals.

Theorem 4.5. 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

g(x) = \overline{g(−x)},
|g^{2}(x)| = \frac{dG(x)}{dG′(x)}.

For every f = (f_{0},f_{1},...) ∈ ExpH_{G}, we define

f′_{n}(x_{1},...,x_{n}) = f_{n}(x_{1},...,x_{n})g(x_{1})··· g(x_{n}), n = 1,2,..., f′_{0}= f_{0}.

Then f′= (f′_{0},f′_{1},...) ∈ ExpH_{G′}^{n}, and

f_{0}+ ∑_{n=1}^{∞} ∫ \frac{1}{n!}f_{n}(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n})

=^{Δ} f′_{0}+ ∑_{n=1}^{∞} \frac{1}{n!} ∫ f′_{n}(x_{1},...,x_{n})Z_{G′}(dx_{1})...Z_{G′}(dx_{n}),

where Z_{G} and Z_{G′} are Gaussian random spectral measures corresponding to G and G′.

Proof of Theorem 4.5. We have ∥f′_{n}∥_{G′}= ∥f_{n}∥_{G}, hence f′∈ ExpH_{G′}. Let φ_{1},φ_{2},... be a complete orthonormal system in H_{G}^{1}. Then φ′_{1},φ′_{2},..., φ′_{j}(x) = φ_{j}(x)g(x) for all j = 1,2,... is a complete orthonormal system in H_{G′}^{1}. All functions f_{n} ∈ H_{G}^{n} can be written in the form f(x_{1},...,x_{n}) = ∑ c_{j_{1},...,j_{n}} Sym(φ_{j_{1}}(x_{1})··· φ_{j_{n}}(x_{n})). Then f′(x_{1},...,x_{n}) = ∑ c_{j_{1},...,j_{n}} Sym(φ′_{j_{1}}(x_{1})··· φ′_{j_{n}}(x_{n})). Rewriting all terms

∫ Sym(φ_{j_{1}}(x_{1})··· φ_{j_{n}}(x_{n}))Z_{G}(dx_{1})...Z_{G}(,dx_{n})

and

∫ Sym(φ′_{j_{1}}(x_{1})··· φ′_{j_{n}}(x_{n}))Z_{G′}(dx_{1})...Z_{G′}(,dx_{n})

by means of Itô’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. ⊓ ⊔

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.

Lemma 4.6. Define for all t > 0 the (multiplication) transformation T_{t}x = tx either from R^{ν} to R^{ν} or from the torus [−π,π)^{ν} to the torus [−tπ,tπ)^{ν}. Given a spectral measure G on R^{ν} or on [−π,π)^{ν} define the spectral measure G_{t} on R^{ν} or on [−tπ,tπ)^{ν} by the formula G_{t}(A) = G(\frac{A}{t}) for all measurable sets A, and similarly define the function f_{k,t}(x_{1},...,x_{k}) = f_{k}(tx_{1},...,tx_{k}) for all measurable functions f_{k} of k variables, k = 1,2,..., with x_{j} ∈ R^{ν} or x_{j} ∈ [−π,π)^{ν} for all 1 ≤ j ≤ k, and put f_{0,t} = f_{0}. If f = (f_{0},f_{1},...) ∈ ExpH_{G}, then f_{t} = (f_{0,t},f_{1,t},...) ∈ ExpH_{G_{t}}, and

f_{0}+ ∑_{n=1}^{∞} ∫ \frac{1}{n!}f_{n}(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n})

=^{Δ} f_{0,t}+ ∑_{n=1}^{∞} \frac{1}{n!} ∫ f_{n,t}(x_{1},...,x_{n})Z_{G_{t}}(dx_{1})...Z_{G_{t}}(dx_{n}),

where Z_{G} and Z_{G_{t}} are Gaussian random spectral measures corresponding to G and G′.

Proof of Lemma 4.6. It is easy to see that f_{t} = (f_{0,t},f_{1,t},...) ∈ ExpH_{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{A}{t}). But with such a choice of Z_{G_{t}} we can write even =

instead of =^{Δ} in this formula. ⊓ ⊔

The next result shows a relation between Wick polynomials and Wiener–Itô integrals.

Theorem 4.7. Let a stationary Gaussian field be given, and let Z_{G} denote the random spectral measure adapted to it. Let P(x_{1},...,x_{m}) = ∑ c_{j_{1},...,j_{n}} x_{j_{1}} ··· x_{j_{n}} be a homogeneous polynomial of degree n, and let h_{1},...,h_{m}∈ H_{G}^{1}. (Here j_{1},...,j_{n} are n indices such that 1 ≤ j_{l} ≤ m for all 1 ≤ l ≤ n. It is possible that j_{l} = j_{l′}also if l ≠ l′.) Define the random variables ξ_{j} = ∫ h_{j}(x)Z_{G}(dx), j = 1,2,...,m, and the function P̃(u_{1},...,u_{n}) = ∑ c_{j_{1},...,j_{n}} h_{j_{1}}(u_{1})··· h_{j_{n}}(u_{n}). Then

: P(ξ_{1},...,ξ_{m}): = ∫ P̃(u_{1},...,u_{n})Z_{G}(du_{1})...Z_{G}(du_{n}).

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,

: P(ξ_{1},...,ξ_{m}): =: P_{1}(ξ_{1},...,ξ_{m}):

Proof of Theorem 4.7. It is enough to show that

: ξ_{j_{1}} ··· ξ_{j_{n}}: = ∫ h_{j_{1}}(u_{1})··· h_{j_{n}}(u_{n})Z_{G}(du_{1})...Z_{G}(du_{n}).

If h_{1},...,h_{m} ∈ H_{G}^{1} are orthonormal, (all functions h_{l} have norm 1, and if l ≠ 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 Itô’s formula. In the general case an orthonormal system h̄_{1},...,h̄_{m}can be found such that

h_{j} = ∑_{k=1}^{m} c_{j,k}h̄_{k}, j = 1,...,m

with some real constants c_{j,k}. Set η_{k} = ∫ h̄_{j}Z_{G}(dx). Then

: ξ_{j_{1}} ··· ξ_{j_{n}}: = : ( ∑_{k=1}^{m} c_{j_{1},k}η_{k}) ··· ( ∑_{k=1}^{m} c_{j_{n},k}η_{k}) :
= ∑_{k_{1},...,k_{n}}c_{j_{1},k_{1}} ··· c_{j_{n},k_{n}}: η_{k_{1}}··· η_{k_{n}}:
= ∑_{k_{1},...,k_{n}}c_{j_{1},k_{1}} ··· c_{j_{n},k_{n}} ∫ h̄_{k_{1}}(u_{1})··· h̄_{k_{n}}(u_{n})Z_{G}(du_{1})...Z_{G}(du_{n})
= ∫ h_{j_{1}}(u_{1})··· h_{j_{n}}(u_{n})Z_{G}(du_{1})...Z_{G}(du_{n})

as we claimed. ⊓ ⊔

We finish this chapter by showing how the Wiener–Itô 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–Itô 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^{ν}, and let Z_{G} be a corresponding Gaussian spectral random measure on a probability space (Ω,A ,P). Let us define a new spectral measure Ĝ= G × λ_{[− \frac{1}{2}, \frac{1}{2}]} on R^{ν+1} , where λ_{[− \frac{1}{2}, \frac{1}{2}]} denotes the uniform distribution

on the interval [−\frac{1}{2}, \frac{1}{2}]. If the probability space (Ω,A ,P) is sufficiently rich, a random spectral measure ZĜ corresponding to Ĝ can be defined on it in such a way that ZĜ(A × [−\frac{1}{2}, \frac{1}{2}]) = Z_{G}(A) for all A ∈ B^{ν}. For f ∈ H̄_{G}^{n} we define the function f̂∈ H̄ n ; Ĝ by the formula f̂(y_{1},...,y_{n}) = f(x_{1},...,x_{n}) if y_{j} is the juxtaposition (x_{j},u_{j}), x_{j} ∈ R^{ν}, u_{j} ∈ R^{1}, j = 1,2,...,n. Finally we define the Wiener–Itô integral in the general case by the formula

∫ f(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n}) = ∫ f̂(y_{1},...,y_{n})ZĜ(dy_{1})...ZĜ(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–Itô integrals coincides with the original one. It is easy to check this fact for one-fold integrals, and then Itô’s formula proves it for multiple integrals. It can be seen with the help of Itô’s formula again, that all results of this chapter remain valid for the new definition of Wiener–Itô 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 ∈ H_{G}^{n} can be written in the form

f(x_{1},...,x_{n}) = ∑ c_{j_{1},...,j_{n}} φ_{j_{1}}(x_{1})··· φ_{j_{n}}(x_{n})  (4.9)

4 Multiple Wiener–Itô integrals 41

with some functions φ_{j} ∈ H_{G}^{1}, j = 1,2,.... Define ξ_{j} = ∫ φ_{j}(x)Z_{G}(dx). If f has the form (4.9), then

∫ f(x_{1},...,x_{n})Z_{G}(dx_{1})...Z_{G}(dx_{n}) = ∑ c_{j_{1},...,j_{n}} : ξ_{j_{1}} ··· ξ_{j_{n}}: .

The last identity would provide another possibility for defining Wiener–Itô integrals.

Chapter 5

The proof of Itô’s formula. The diagram formula and some of its consequences

We shall prove Itô’s formula with the help of the following Proposition 5.1. Let f ∈ H̄_{G}^{n} and h ∈ H̄_{G}^{1} . Let us define the functions

f ×_{k} h(x_{1},...,x_{k−1},x_{k+1},...,x_{n}) = ∫ f(x_{1},...,x_{n})\overline{h(x_{k})}G(dx_{k}), k = 1,...,n,

and

fh(x_{1},...,x_{n+1} ) = f(x_{1},...,x_{n})h(x_{n+1} ).

Then f ×_{k} h, k = 1,...,n, and fh are in H̄_{G}^{n−1} and H̄_{G}^{n+1} respectively, and their norm satisfies the inequality ∥f ×_{k} h∥ ≤ ∥f∥·∥h∥ and ∥fh∥ ≤ ∥f∥·∥h∥. The relation

n!I_{G}(f)I_{G}(h) = (n + 1)!I_{G}(fh) + ∑_{k=1}^{n}(n − 1)!I_{G}(f ×_{k} h)

holds true.

We shall get Proposition 5.1 as the special case of the diagram formula formulated in Theorem 5.3. Remark. There is a small inaccuracy in the formulation of Lemma 5.1. We considered the Wiener–Itô integral of the function f ×_{k} h with arguments x_{1},…, x_{k−1}, x_{k+1},…, x_{n}, while we defined this integral for functions with arguments x_{1},...,x_{n−1} . We can correct this inaccuracy for instance by reindexing the variables of f ×_{k} h and working with the function

(f ×_{k} h)′(x_{1},...,x_{n−1} ) = f ×_{k} h(x_{α_{k}(1)},...,x_{α_{k}(k−1)} ,x_{α_{k}(k+1)} ,...,x_{α_{k}(n)})

instead of f ×_{k} h, where α_{k}(j) = j for 1 ≤ j ≤ k − 1, and α_{k}(j) = j − 1 for k + 1 ≤ j ≤ n.

We also need the following recursion formula for Hermite polynomials.

Lemma 5.2. The identity

H_{n}(x) = xH_{n−1} (x) − (n − 1)H_{n−2} (x) for n = 1,2,...,

holds with the notation H_{−1} (x) ≡ 0.

Proof of Lemma 5.2.

H_{n}(x) = (−1)^{n}e^{x^{2}/2} \frac{d^{n}}{dx^{n}}(e^{−x^{2}/2}) = −e^{x^{2}/2} \frac{d}{dx}( H_{n−1} (x)e^{−x^{2}/2})
= x H_{n−1} (x) − \frac{d}{dx}H_{n−1} (x).

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) + ∑_{j=0}^{n−3} c_{j}H_{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

∫ e^{−x^{2}/2}H_{j}(x)\frac{d}{dx}H_{n−1} (x)dx= −∫ H_{n−1} (x)\frac{d}{dx}(e^{−x^{2}/2}H_{j}(x))dx
= ∫ e^{−x^{2}/2}H_{n−1} (x)P_{j+1} (x)dx = 0

with the polynomial P_{j+1} (x) = xH_{j}(x) − \frac{d}{dx}H_{j}(x) of order j + 1 for j ≤ n − 3. ⊓ ⊔

Proof of Theorem 4.3 via Proposition 5.1. We prove Theorem 4.3 by induction. Theorem 4.3 holds for N = 1. Assume that it holds for N − 1. Let us define the functions

f(x_{1},...,x_{N−1}) = g_{1}(x_{1})··· g_{N−1}(x_{N−1})
h(x)= g_{N}(x).

Then

J = ∫ g_{1}(x_{1})··· g_{N}(x_{N})Z_{G}(dx_{1})...Z_{G}(dx_{N})
= N!I_{G}(fh) = (N − 1)!I_{G}(f)I_{G}(h) − ∑_{k=1}^{N−1}(N − 2)!I_{G}(f ×_{k} h)

by Proposition 5.1. We can show with the help of our induction hypothesis that

J = H_{j_{1}} ( ∫ φ_{1}(x)Z_{G}(dx)) ··· H_{j_{m−1}} ( ∫ φ_{m−1}(x)Z_{G}(dx))
H_{j_{m}−1} ( ∫ φ_{m}(x)Z_{G}(dx)) ∫ φ_{m}(x)Z_{G}(dx)
−(j_{m}− 1)H_{j_{1}} ( ∫ φ_{1}(x)Z_{G}(dx)) ··· H_{j_{m−1}} ( ∫ φ_{m−1}(x)Z_{G}(dx))
H_{j_{m}−2} ( ∫ φ_{m}(x)Z_{G}(dx)) ,

where H_{j_{m}−2} (x) = H_{−1} (x) ≡ 0 if j_{m}= 1. This relation holds, since

f ×_{k} h(x_{1},...,x_{k−1},x_{k+1},...,x_{N−1}) = ∫ g_{1}(x_{1})··· g_{N−1}(x_{N−1})\overline{φ_{m}(x_{k})}G(dx_{k})

= \{ \begin{matrix}0 if k ≤ N − j_{m} \\ g_{1}(x_{1})··· g_{k−1}(x_{k−1})g_{k+1}(x_{k+1})··· g_{N−1}(x_{N−1}) if N − j_{m}< k ≤ N − 1. \end{matrix}

Hence Lemma 5.2 implies that

J = ∏_{s=1}^{m−1} H_{j_{s}} ( ∫ φ_{s}(x)Z_{G}(dx)) [H_{j_{m}−1} ( ∫ φ_{m}(x)Z_{G}(dx)) ∫ φ_{m}(x)Z_{G}(dx)

−(j_{m}− 1)H_{j_{m}−2} ( ∫ φ_{m}(x)Z_{G}(dx))] = ∏_{s=1}^{m} H_{j_{s}} ( ∫ φ_{s}(x)Z_{G}(dx)) ,

as claimed. ⊓ ⊔

Let us fix some functions h_{1} ∈ H̄_{G}^{n_{1}} ,…, h_{m}∈ H̄_{G}^{n_{m}}. In the next result, in the socalled diagram formula, we express the product n_{1}!I_{G}(h_{1})··· n_{m}!I_{G}(h_{m}) as the sum of Wiener–Itô 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 [7].

We shall use the term diagram of order (n_{1},...,n_{m}) for an undirected graph of n_{1}+··· +n_{m} vertices such that its vertices are indexed by the pairs of integers (j,l), l = 1,...,m, j = 1,...,n_{l}, with the properties that no more than one edge enters into each vertex, and edges can connect only pairs of vertices (j_{1},l_{1}) and (j_{2},l_{2}) for which l_{1} ≠ l_{2}. Let Γ = Γ (n_{1},...,n_{m}) denote the set of all diagrams. Given a diagram γ ∈ Γ |γ| denotes the number of edges in γ. Let there be given a set of functions h_{1} ∈ H̄_{G}^{n_{1}} ,…, h_{m} ∈ H̄_{G}^{n_{m}}. Sometimes we denote the variables of the function h_{l} by x_{(j,l)} instead of x_{j}, i.e. we write h_{l}(x_{(1,l)},...,x_{(n_{l},l)}) instead of h_{l}(x_{1},...,x_{n_{l}}). Put N = n_{1} + ··· + n_{m}. We introduce the function of N variables corresponding to the vertices of the diagram by the formula

h(x_{(j,l)}, l = 1,...,m, j = 1,...,n_{l}) = ∏_{l=1}^{m} h_{l}(x_{(j,l)}, j = 1,...,n_{l}).

Fixing a diagram γ ∈ Γ 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,...,N − 2|γ| and the vertices connected by an edge will have the numbers p and p + |γ|, where p = N − 2|γ| + 1,...,N − |γ|. In such a way we have defined the function h(x_{1},...,x_{N}) (with an enumeration of the indices of the variables depending on the diagram γ). After this definition of the function h(x_{1},...,x_{N}) we take that function of N −|γ| variables which we get by replacing the arguments x_{N−|γ|+p} by the arguments −x_{N−2|γ|+p} , 1 ≤ p ≤ |γ. Then we get the function h_{γ} appearing in the diagram formula by integrating

this function by the product measure ∏_{p=1}^{|γ|} G(dx_{N−2|γ|+p} ).

More explicitly, we write

h_{γ}(x_{1},...,x_{N−2|γ|}) = ∫ ··· ∫ h(x_{1},...,x_{N−|γ|},−x_{N−2|γ|+1} ,...,−x_{N−|γ|})
G(dx_{N−2|γ|+1} )...G(dx_{N−|γ|}).  (5.1)

The function h_{γ} depends only on the variables x_{1},...,x_{N−2|γ|}, 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 + γ of two vertices connected by an edge does not change the value of the integral h_{γ}. Let us now consider I_{G}(h_{γ}). The function h_{γ} may depend on the numbering of those vertices of γ from which no edge starts, but Symh_{γ} and therefore I_{G}(h_{γ}) does not depend on it. Now we formulate the following

Theorem 5.3. (Diagram formula.) For all functions h_{1} ∈ H̄_{G}^{n_{1}} ,…, h_{m} ∈ H̄_{G}^{n_{m}}, n_{1},...,n_{m}= 1,2,..., the following relations hold:

(A) h_{γ} ∈ H̄_{G}^{n−2|γ|} , and ∥h_{γ}∥ ≤ ∏_{j=1}^{m} ∥h_{j}∥ for all γ ∈ Γ .

(B) n_{1}!I_{G}(h_{1})··· n_{m}!I_{G}(h_{m}) = ∑_{γ∈Γ} (N − 2|γ|)!I_{G}(h_{γ}).

Remark. 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 ∈ H̄_{G}^{1} .

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},...,x_{n+m} instead of x_{(1,1)},...,x_{(n,1)},x_{(1,2)}...,x_{(m,2)}. It is clear that the function h_{γ} satisfies Property (a) of the classes H̄_{G}^{j} defined in Chapter 4. We show that Part (A) of Theorem 5.3 is a consequence of the Schwartz inequality. To prove this estimate on the norm of h_{γ} it is enough to restrict ourselves to such diagrams γ in which the vertices (n,1) and (m,2), (n−1,1) and (m−1,2),…, (n−k,1) and (m−k,2) are connected by edges with some 0 ≤ k ≤ min(n,m). In this case we can write

|h_{γ}(x_{1},...,x_{n−k−1} ,x_{n+1} ,...,x_{n+m−k−1} )|^{2}
= |∫ h_{1}(x_{1},...,x_{n})h_{2}(x_{n+1} ,...,x_{n+m−k−1} ,−x_{n−k},...,−x_{n})

G(dx_{n−k})...G(dx_{n})|^{2}
≤ ∫ |h_{1}(x_{1},...,x_{n})|^{2}G(dx_{n−k})...G(dx_{n})
∫ |h_{2}(x_{n+1} ,...,x_{n+m})|^{2}G(dx_{n+m−k} )...G(dx_{n+m})

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} ∈ ˆ ¯H_{G}^{n} and h_{2} ∈ ˆ ¯H_{G}^{m}. Assume that they are adapted to a regular system D = {Δ_{j}, j = ±1,...,±N} of subsets of R^{n} with finite measure G. We may even assume that all Δ_{j} ∈ D satisfy the inequality G(Δ_{j}) < ε with some ε > 0 to be chosen later, because otherwise we could split up the sets Δ_{j} into smaller ones. Let us fix a point u_{j} ∈ Δ_{j} in all sets Δ_{j} ∈ D. Put K_{i} = sup_{x}|h_{i}(x)|, i = 1,2, and let A be a cube containing all Δ_{j}.

We can write

I = n!I_{G}(h_{1})m!I_{G}(h_{2}) = ∑ ′h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{1}},...,u_{k_{m}})
Z_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})Z_{G}(Δ_{k_{1}})··· Z_{G}(Δ_{k_{m}})

with the numbers u_{j_{p}} ∈ Δ_{j_{p}} and u_{k_{r}} ∈ Δ_{k_{r}} we have fixed, where the summation in ∑′ goes through all pairs ((j_{1},...,j_{n}),(k_{1},...,k_{m})), j_{p}, k_{r} ∈ {±1,...,±N}, p = 1,...,n, r = 1,...,m, such that j_{p} ≠ ±jp̄ and k_{r} ≠ ±kr̄ if p ≠ p̄or r ≠ r̄.

Write

I = ∑_{γ∈Γ} ∑ ^{γ}h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{1}},...,u_{k_{m}})
Z_{G}(Δ_{j_{1}})··· Z_{G}(Δ_{j_{n}})Z_{G}(Δ_{k_{1}})··· Z_{G}(Δ_{k_{m}}),

where ∑^{γ} contains those terms of ∑′for which j_{p} = k_{r} or j_{p} = −k_{r} if the vertices (1,p) and (2,r) are connected in γ, and j_{p} ≠ ±k_{r} if (1,p) and (2,r) are not connected. Let us define the sets

A_{1} = A_{1}(γ) = {p: p ∈ {1,...,n}, and no edge starts from (p,1) in γ},
A_{2} = A_{2}(γ) = {r: r ∈ {1,...,m}, and no edge starts from (r,2) in γ}

and

B = B(γ) = {(p,r): p ∈ {1,...,n}, r ∈ {1,...,m},

(p,1) and (r,2) are connected in γ} together with the map α: {1,...,n} \ A_{1} → {1,...,m} \ A_{2} defined as

α(p) = r if (p,r) ∈ B for all p ∈ {1,...,n} \ A_{1}.  (5.2)

Let Σ^{γ} denote the value of the inner sum ∑^{γ} for some γ ∈ Γ in the last summation formula, and write it in the form

Σ^{γ} = Σ_{1}^{γ}+ Σ_{2}^{γ}

with

Σ_{1}^{γ} = ∑ ^{γ}h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{1}},...,u_{k_{m}}) ∏_{p∈A_{1}}Z_{G}(Δ_{j_{p}}) ∏_{r∈A_{2}}Z_{G}(Δ_{k_{r}})

· ∏_{(p,r)∈B}E (Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}))

and

Σ_{2}^{γ} = ∑ ^{γ}h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{1}},...,u_{k_{m}}) ∏_{p∈A_{1}}Z_{G}(Δ_{j_{p}}) ∏_{r∈A_{2}}Z_{G}(Δ_{k_{r}})

·[ ∏_{(p,r)∈B}Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) − E ( ∏_{(p,r)∈B}Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}})) ] .

The random variables Σ_{1}^{γ} and Σ_{2}^{γ} are real valued. To see this observe that if the sum defining these expressions contains a term with arguments Δ_{j_{p}}, and Δ_{k_{r}}, then it also contains the term with arguments −Δ_{j_{p}} and −Δ_{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 Σ_{1}^{γ} = \overline{Σ_{1}^{γ}} and Σ_{2}^{γ} = \overline{Σ_{2}^{γ}}. Hence these random variables are real valued. As a consequence, we can bound (n+m−2|γ|)!I_{G}(h_{γ})−Σ_{1}^{γ} and Σ_{2}^{γ} by means of their second moment.

We are going to show that Σ_{1}^{γ} is a good approximation of (n + m − 2|γ|)!I_{G}(h_{γ}), and Σ_{2}^{γ} is negligibly small. This implies that (n + m − 2|γ|)!I_{G}(h_{γ}) well approximates Σ^{γ}. 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|γ|)!I_{G}(h_{γ}) − Σ_{1}^{γ} we rewrite Σ_{1}^{γ} as a Wiener–Itô integral with a kernel function adapted to the regular system D which is close to h_{γ}. To find this kernel function we rewrite the sum defining Σ_{1}^{γ} by first fixing the variables u_{j_{p}}, p ∈ A_{1}, and u_{k_{r}}, r ∈ A_{2}, and summing up by the remaining variables, and after this summing by the variables fixed at the first step. We get that

Σ_{1}^{γ} = ∑_{j_{p}: 1≤|j_{p}|≤N for all p∈A_{1} k_{r}: 1≤|k_{r}|≤N for all r∈A_{2}} h_{γ,1}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2})

∏_{p∈A_{1}}Z_{G}(Δ_{j_{p}}) ∏_{r∈A_{2}}Z_{G}(Δ_{k_{r}})  (5.3)

with a function h_{γ,1} depending on the arguments j_{p}, p ∈ A_{1}, and k_{r}, r ∈ A_{2}, with values j_{p},k_{r} ∈ {±1,...,±N} defined with the help another function h_{γ,2} described below. It also depends on the arguments j_{p}, p ∈ A_{1}, and k_{r}, r ∈ A_{2}, with values j_{p},k_{r} ∈ {±1,...,±N}. More explicitly, formula (5.3) holds with the function h_{γ,1} defined as

h_{γ,1}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2}) = 0  (5.4)

if the numbers in the set {±j_{p}: p ∈ A_{1}} ∪ {±k_{r}: r ∈ A_{2}} are not all different, and

h_{γ,1}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2}) = h_{γ,2}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2})  (5.5)

if all numbers ±j_{p}, p ∈ A_{1}, and ±k_{r}, r ∈ A_{2} are different, where we define the function h_{γ,2}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2}) for all sequences j_{p}, p ∈ A_{1} and k_{r}, r ∈ A_{2}, with j_{p},k_{r} ∈ {±1,...,±N} (i.e. also in the case when some of the arguments j_{p}, p ∈ A_{1}, or k_{r}, r ∈ A_{2}, agree) by the formula

h_{γ,2}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2}) = ∑ ^{γ,1}h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{1}},...,u_{k_{m}})
· ∏_{(p,r)∈B}E (Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}})) .  (5.6)

The value of the sum ∑^{γ,1} in formula (5.6) which depends on the arguments j_{p}, p ∈ A_{1}, and k_{r}, r ∈ A_{2}, is defined in the following way. We sum up for such sequences (j_{1},...,j_{n}) and (k_{1},...,k_{m}) whose coordinates with p ∈ A_{1} and q ∈ A_{2} are fixed, and whose coordinates with indices p ∈ {1,...,n} \ A_{1} and r ∈ {1,...,m} \ A_{2} satisfy the following conditions. Put C = {±j_{p}, p ∈ A_{1}} ∪ {±k_{r}, r ∈ A_{2}}. We demand that all numbers j_{p} and k_{r} with indices p ∈ {1,...,n} \ A_{1} and r ∈ {1,...,m} \ A_{2} are such that j_{p},k_{r} ∈ {±1,...,±N} \ C. To formulate the remaining conditions let us write all numbers r ∈ {1,...,m} \ A_{2} in the form r = α(p), p ∈ {1,...,n} \ A_{1} with the map α defined in (5.2). We also demand that only such sequences appear in the summation whose coordinates k_{r} = k_{α(p)} satisfy the condition k_{α(p)} = ±j_{p} for all p ∈ {1,...,n}\A_{1}. Beside this, all numbers ±j_{p}, p ∈ {1,...,n}\A_{1}, must be different. The summation in ∑^{γ,1} is taken for all such sequences j_{p}, p ∈ {1,...,n} and k_{r}, r ∈ {1,...,m}, whose coordinates with p ∈ {1,...,n} \ A_{1} and r ∈ {1,...,m} \ A_{2} satisfy the above conditions.

Formula (5.6) can be rewritten in a simpler form. To do this let us first observe that the condition k_{α(p)} = ±j_{p} can be replaced by the condition k_{α(p)} = −j_{p} in it, and we can write G(Δ_{j_{p}}) instead of the term EZ_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) (with (p,r) ∈ B) in the product at the end of (5.6). This follows from the fact that EZ_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) = EZ_{G}(Δ_{j_{p}})^{2} = 0 if k_{r} = j_{p} and EZ_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) = EZ_{G}(Δ_{j_{p}} Z_{G}(−Δ_{j_{p}}) = G(Δ_{j_{p}}) if k_{r} = −j_{p}. Beside this, the expression in (5.6) does not change if we take summation for all such sequences for which the number j_{p} with coordinate p ∈ {1,...,n} \ A takes all possible values j_{p} ∈ {±1,...,±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 D, hence h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{1}},...,u_{k_{m}}) = 0 if for an index p ∈ {1,...,n} \ A_{1} j_{p} = ±j_{p′} with p ≠ p′or j_{p} = −k_{r} with (p,r) ∈ B, and beside this there exists some r′∈ A_{2} such that j_{p} = ±k_{r′}.

The above relations enable us to rewrite (5.6) in the following way. Let us define that map α^{−1} on the set {1,...,m} \ A_{2} which is the inverse of the map α defined in (5.2), i.e. α^{−1} (r) = p if (p,r) ∈ B. With this notation we can write

h_{γ,2}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2})

= ∑_{j_{p},p∈{1,...,n}\A_{1}, 1≤|j_{p}|≤N for all indices p} h_{1}(u_{j_{1}},...,u_{j_{n}})h_{2}(u_{k_{r}}, r ∈ A_{2},−u_{j_{α^{−1} (r)}} , r ∈ {1,...,m} \ A_{2})

∏_{p∈{1,...,n}\A_{1}}G(Δ_{j_{p}}).  (5.7)

Formula (5.7) can be rewritten as

h_{γ,2}(j_{p}, p ∈ A_{1}, k_{r}, r ∈ A_{2})  (5.8)
= ∫ h_{1}(u_{j_{p}} , p ∈ A_{1}, x_{p}, p ∈ {1,...,n} \ A_{1})

h_{2}(u_{k_{r}}, r ∈ A_{2}, −x_{α^{−1} (r)}, r ∈ {1,...,m} \ A_{2}) ∏_{p∈{1,...,n}\A_{1}}G(dx_{p}).

We define with the help of h_{γ,1} and h_{γ,2} two new functions on R^{(n+m−2|γ|)ν} with arguments x_{1},...,x_{n+m−2|γ} . The first one will be the kernel function of the Wiener–Itô integral expressing Σ_{1}^{γ}, and the second one will be equal to the function h_{γ} defined in (5.1). We define these functions in two steps. In the first step we reindex the arguments of the functions h_{1,γ} and h_{2},γ to get functions depending on sequences j_{1},...,j_{n+m−2|γ|} . For this goal we list the elements of the sets A_{1} and A_{2} as A_{1} = {p_{1},...,p_{n−|γ|}} with 1 ≤ p_{1} < p_{2} < ···< p_{n−|γ|} ≤ n and A_{2} = {r_{1},...,r_{m−|γ|}} with 1 ≤ r_{1} < r_{2} < ···< r_{m−|γ|} ≤ m, and define the maps β_{1}: A_{1} → {1,...,n − |γ|} and β_{2}: A_{2} → {n − |γ| + 1,...,n + m − 2|γ|} by the formulas β_{1}(p_{l}) = l if 1 ≤ l ≤ n − γ, 1 ≤ l ≤ n − |γ|, and β_{2}(r_{l}) = l + n − |γ|, 1 ≤ l ≤ m − |γ|, if n − |γ| + 1 ≤ l ≤ n + m − 2|γ|. We define with the help of the maps β_{1} and β_{2} the functions

h_{γ,3}(j_{1},...,j_{n+m−2|γ|} ) = h_{γ,1}(j_{β_{1}(r_{1})},...,j_{β_{1}(n−|γ|))},k_{β_{2}(1)},...,k_{β_{2}(m−|γ|)})

and

h_{γ,4}(j_{1},...,j_{n+m−2|γ|} ) = h_{γ,2}(j_{β_{1}(r_{1})},...,j_{β_{1}(n−|γ|))},k_{β_{2}(1)},...,k_{β_{2}(m−|γ|)}),

where the arguments of the functions h_{γ,3} and h_{γ,4} are sequences j_{1},...,j_{n+m−2|γ|} with j_{s} ∈ {±1,...,±N} for all 1 ≤ s ≤ n + m − 2|γ|.

With the help of the above functions we define the following functions h_{γ,5} and h_{γ,6} on R^{(n+m−2|γ|)ν} .

h_{γ,5}(x_{1},...,x_{n+m−2|γ|} ) = \{ \begin{matrix}h_{γ,3}(j_{1},...,j_{n+m−2|γ|} ) if x_{l} ∈ Δ_{j_{l}} , \\ for all 1 ≤ l ≤ n + m − 2|γ| \\ 0 otherwise, \end{matrix}

and

h_{γ,6}(x_{1},...,x_{n+m−2|γ|} ) = \{ \begin{matrix}h_{γ,4}(j_{1},...,j_{n+m−2|γ|} ) if x_{l} ∈ Δ_{j_{l}} , \\ for all 1 ≤ l ≤ n + m − 2|γ| \\ 0 otherwise. \end{matrix}

It follows from relation (5.4) and the definition of the function h_{γ,5} (with the help of the definition of the functions h_{γ,1} and h_{γ,3}) that h_{γ,5} ∈ ˆ ¯H_{G}^{n}, and it is adapted to the regular system D. Then relations (5.3) and the definition of h_{γ,5} also imply that Σ_{1}^{γ} = (n + m − 2|γ|)!I_{G}(h_{γ,5}).

On the other hand, I claim that the function h_{γ} defined in (5.1) satisfies the identity h_{γ} = h_{γ,6}. Actually a slightly weaker statement would suffice for us. It would be enough to show that the function h_{γ} can be obtained by a permutation of the indices of the variables in h_{γ,6}, because the value of a Wiener-Itô integral does not change by a reindexation of the variables of the kernel function in it.

To prove the desired identity let us write up the expression defining the function h_{γ} in the present case. First we define the function h(x_{1},...,x_{n+m}) (depending on the diagram γ) which is applied in the definition of h_{γ}. To do this we introduce some notations.

Put A_{1} = {j_{1},...,j_{n−|γ|}}, 1 ≤ j_{1} < j_{2} < ··· < j_{n−|γ|}, {1,...,n}\A_{1} = {l_{1},...,l_{|γ|}}, 1 ≤ l_{1} < l_{2} < ···< l_{m−|γ|}, and A_{2} = {j′_{1},...,j′_{m−|γ|}}, 1 ≤ j′_{1} < j′_{2} < ···< j′_{|γ|}, {1,...,m}\A_{2} = {l′_{1},...,l′_{m−|γ|}}, 1 ≤ l′_{1}< l′_{2}< ··· < l′_{m−|γ|}, and define with their help the following functions. Define the function β(·) on the set {1,...,n} as β(k) = p if k = j_{p} ∈ A_{1}, and β(k) = n+m−2|γ|+q if k = l_{q} ∈ {1,...,n}\A_{1}. Define similarly the function δ(·) on the set {n + 1,...,n + m} as δ(k) = p + |γ| if k − n = j′_{p} ∈ A_{2}. If k − n = l′_{q} ∈ {1,...,m} \ A_{2}, then there is an edge (l_{p},l′_{q}) ∈ B of the diagram γ, and we define δ(k) = n − |γ| + p with the index p of the number l_{p} in this case.

With the help of the above notations we can define the function h(x_{1},...,x_{n+m}) which takes part in the definition of h_{γ} in formula (5.1) as

h(x_{1},...,x_{n+m}) = h_{1}(x_{β(1)},...,x_{β(n)})h_{2}(x_{δ(n+1)} ,...,x_{δ(n+m)}).

To define the kernel function of the integral in (5.1) observe that the set {δ(n + 1),...,δ(n + m)} agrees with the set {n − |γ| + 1,...,n + m − 2|γ|} ∪ {n − |γ| + 1,...,n + m} = C_{1} ∪ C_{2}. Put δ̄(k) = δ(k) if δ(k) ∈ C_{1} and δ̄(k) = δ(k) − |γ| if δ(k) ∈ C_{2}. Let us also introduce ε(j) = 1 if n − |γ + 1 ≤ j ≤ n + m − 2|γ|, and ε(j) = −1 if n + m − 2|γ| + 1 ≤ j ≤ n + m − |γ|. With such a notation we can write

h(x_{1},...,x_{n+m−|γ|} ,−x_{n+m−2|γ|+1} ,...,−x_{n+m−|γ|} )
= h_{1}(x_{β(1)},...,x_{β(n)})h_{2}(ε(δ̄(n + 1))x_{δ̄(n+1)} ,...,ε(δ̄(n + m))x_{δ̄(n+m)})

as the kernel function in the integral (5.1) defining the function h_{γ}(x_{1},...,x_{n+m−2|γ|} ) in the present case.

By formula (5.1) we can calculate the function h_{γ}(x_{1},...,x_{n+m−2|γ|} ) by integrating the above defined function h(x_{1},...,x_{n+m−|γ|} ,−x_{n+m−2|γ|+1} ,...,−x_{n+m−|γ|} ) with respect to the measure G(dx_{N−2|γ|+1} )...G(dx_{N−|γ|}). By comparing this formula with the definition of the function h_{γ,2} defined in (5.8) together with the definition of the functions h_{γ,4} and h_{γ,6} with its help one can see that the identity h_{γ,6}= h_{γ} holds.

Observe that the function h_{γ} disappears also in such points (x_{1},...,x_{n+m−2|γ|} ) for which x_{l} ∈ Δ_{j_{l}} for all 1 ≤ l ≤ n + m − 2|γ| with such indices j_{l} for which some of the numbers in the set {±j_{1},...,±j_{n−|γ|}} or in the set {±j_{n−|γ|+1} ,...,±j_{n+m−2|γ|} } agree. This fact together with the identity h_{γ} = h_{γ,6} and the relation between the functions h_{γ,5} and h_{γ,6} (implied by the definition of the function h_{γ,1} in formulas (5.4) and (5.5)) yield the identity

h_{γ}(x_{1},...,x_{n+m−2|γ|} ) = h_{γ,5}(x_{1},...,x_{n+m−2|γ|} ) + h_{γ,7}(x_{1},...,x_{n+m−2|γ|} )

with

h_{γ,7}(x_{1},...,x_{n+m−2|γ|} )

= \{ \begin{matrix}h_{γ}(x_{1},...,x_{n+m−2|γ|} ) if there exist indices j_{l}, 1 ≤ |j_{l}| ≤ N, \\ 1 ≤ l ≤ n + m − 2|γ| such that x_{l} ∈ Δ_{j_{l}} , 1 ≤ l ≤ n + m − 2|γ|, \\ all numbers ± j_{1},...,±j_{n−2|γ|} are different, \\ all numbers ± j_{n−|γ|+1} ,...,±j_{n+m−2|γ|} are different, \\ and \{±j_{1},...,±j_{n−|γ|}\} ∩ \{±j_{n−|γ|+1} ,...,±j_{n+m−2|γ|} \} ≠ ∅ \\ 0 otherwise. \end{matrix}

Since Σ_{1}^{γ} = (n + m − 2|γ|)!I_{G}(h_{γ,5}), we have

(n + m − 2|γ|)!I_{G}(h_{γ}) − Σ_{1}^{γ} = (n + m − 2|γ|)!I_{G}(h_{γ,7}),

and

E(Σ_{1}^{γ}− (n + m − 2|γ|)!I_{G}(h_{γ}))^{2} ≤ (n + m − 2|γ|)!∥(h_{γ,7})∥^{2}

with the norm ∥ · ∥ in H̄_{G}^{n+m−2|γ|} .

On the other hand,

sup|h_{γ,7}(x_{1},...,x_{n+m−2|γ|} )| ≤ sup|h_{γ}(x_{1},...,x_{n+m−2|γ|} )| ≤ K_{1}K_{2}L^{|γ|},

with K_{1} = sup|h_{1}|, K_{2} = sup|h_{2}|, and L = G(A), where A is a fixed cube containing all Δ_{j}. Hence

E(Σ_{1}^{γ}− (n + m − 2|γ|)!I_{G}(h_{γ}))^{2} ≤ C_{1}∥(h_{γ,7})∥^{2}
≤ C_{2}∑ ′′G(Δ_{j_{1}})··· G(Δ_{j_{n+m−2|γ|}} )
≤ C sup_{j}G(Δ_{j}) ≤ Cε,  (5.9)

where the summation ∑′′goes for such sequences j_{1},...,j_{n+m−2|γ|} , 1 ≤ |j_{l}| ≤ N for all 1 ≤ l ≤ n + m − 2|γ|, for which all numbers ±j_{1},...,±j_{n−|γ|} are different, the same relation holds for the elements of the sequence ±j_{n−|γ|+1} ,...,±j_{n+m−2|γ|} , and

{±j_{1},...,±j_{n−|γ|}} ∩ {±j_{n−|γ|+1} ,...,±j_{n+m−2|γ|} } ≠ ∅.

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 D, hence in particular on the parameter ε. In the verification of (5.9) we can exploit that each term in the sum ∑′′is a product which contains a factor G(Δ_{j})^{2} ≤ εG(Δ_{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 Statement B.

Now we turn to the estimation of E(Σ_{2}^{γ})^{2}. It can be expressed as a linear combination of terms of the form

Σ_{3}^{γ}(j_{p},k_{r},jp̄,kr̄, p,p̄∈ {1,...,n}, r,r̄∈ {1,...,m})  (5.10)

= E( ( ∏_{p∈A_{1}}Z_{G}(Δ_{j_{p}}) ∏_{r∈A_{2}}Z_{G}(Δ_{k_{r}}) ∏_{p̄∈A_{1}}Z_{G}(Δ_{jp̄}) ∏_{r̄∈A_{2}}Z_{G}(Δ_{kr̄}))

[ ∏_{(p,r)∈B}Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) − E ∏_{(p,r)∈B}Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}})]

[ ∏_{(p̄,r̄)∈B}Z_{G}(Δ_{jp̄})Z_{G}(Δ_{kr̄}) − E ∏_{(p̄,r̄)∈B}Z_{G}(Δ_{jp̄})Z_{G}(Δ_{kr̄})] ) ,

where Σ_{3}^{γ} depends on such sequences of numbers j_{p}, k_{r}, jp̄, kr̄ with indices 1 ≤ p,p̄≤ n and 1 ≤ r,r̄≤ m for which j_{p},k_{r},jp̄,kr̄∈ {±1,...,±N} for all indices p,r,p̄ and r̄, j_{p} = k_{r} or j_{p} = −k_{r} if (p,r) ∈ B, otherwise all numbers ±j_{p}, ±k_{r} are different, and the same relations hold for the indices jp̄and kr̄if p is replaced by p̄and r is replaced by r̄. 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}, jp̄, kr̄) the expression Σ_{3}^{γ} equals zero, and it is also small in the remaining cases.

Let us fix a sequence of arguments j_{p}, k_{r}, jp̄, kr̄ of Σ_{3}^{γ}, and let us estimate its value with these arguments. Define the sets

A = {j_{p}: p ∈ A_{1}} ∪ {k_{r}: r ∈ A_{2}} and Ā= {jp̄: p̄∈ A_{1}} ∪ {kr̄: r̄∈ A_{2}}.

We claim that Σ_{3}^{γ} equals zero if Ā ≠ −A. In this case there exists an index l ∈ A such that −l ∉ Ā. Let us carry out the multiplication in (5.10). 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}(Δ_{l}) has zero expectation. To see this observe that the set Δ_{l} appears exactly once among the arguments of the terms Z_{G}(Δ_{j_{p}}) and Z_{G}(Δ_{k_{r}}), and none of these terms contains the argument −Δ_{l} = Δ_{−l} . Although −l ∉ Ā, it may happen that l ∈ Ā. In this case the product under investigation contains the independent factor Z_{G}(Δ_{l})^{2} with EZ_{G}(Δ_{l})^{2} = 0. If l ∉ Ā, then there are two possibilities. Either this product contains an independent factor of the form Z_{G}(Δ_{l}) with EZ_{G}(Δ_{l}) = 0, or there is a pair (p̄,r̄) ∈ B such that (jp̄,kr̄) = (±l,±l), and an independent factor of the form Z_{G}(Δ_{l})Z_{G}(±Δ_{−l} )Z_{G}(±Δ_{l}) with the property EZ_{G}(Δ_{l})Z_{G}(±Δ_{−l} )Z_{G}(±Δ_{l}) = 0 appears. Hence Σ_{3}^{γ} = 0 in this case.

Let

F = ⋃_{(p,r)∈B}{j_{p},k_{r}} and F̄ = ⋃_{(p̄,r̄)∈B}{(jp̄,kr̄}.

A factorization argument shows again that the expression in (5.10) equals zero if the sets F ∪ (−F) and F̄ ∪ (−F̄) are disjoint. We can restrict ourselves to the case A = −Ā, and in this case ±A is disjoint both of F ∪ (−F) and F̄ ∪ (−F̄), and the product under investigation contains the independent factor

∏_{(p,r)∈B}Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) − E ∏_{(p,r)∈B}Z_{G}(Δ_{j_{p}})Z_{G}(Δ_{k_{r}}) with expectation zero.

Moreover, if F ∪ (−F) and F̄∪ (−F̄) are not disjoint, (and A = −Ā), then the absolute value of the expression in (5.10) can be estimated from above by

C ε ∏ G(Δ_{j_{p}})G(Δ_{k_{r}})G(Δ_{jp̄})G(Δ_{kr̄})  (5.11)

with a universal constant C < ∞ depending only on the parameters n and m, where the indices j_{p}, k_{r}, jp̄, kr̄ are the same as in (5.10) with the following difference: All indices appear in (5.11) with multiplicity 1, and if both indices l and −l are present in (5.10), then one of them is omitted form (5.11). Moreover, for all j_{p}, k_{r}, jp̄ and kr̄ one of terms G(Δ_{±j_{p}} ), G(Δ_{±k_{r}} ), G(Δ_{±jp̄} ) and G(Δ_{±kr̄}) really appears in this product. The multiplying term ε appears in (5.11), since by carrying out the multiplications in (5.10) and factorizing each term, we get that all non-zero terms have a factor either of the form

EZ_{G}(Δ)^{2}Z_{G}(−Δ)^{2} = E(ReZ_{G}(Δ)^{2}+ ImZ_{G}(Δ)^{2})^{2}
= E ReZ_{G}(Δ)^{4}+ E ImZ_{G}(Δ)^{4}+ 2E ReZ_{G}(Δ)^{2}E ImZ_{G}(Δ)^{2} = 8G(Δ)^{2}

or of the form (E|Z_{G}(Δ)|^{2})^{2} = G(Δ)^{2}, and G(Δ) < ε for all Δ ∈ D. (We did not mention the possibility of an independent factor of the form EZ_{G}(Δ)^{4} or EZ_{G}(Δ)^{3}Z_{G}(−Δ) with Δ ∈ D, because as some calculation shows, EZ_{G}(Δ)^{4} = 0 and EZ_{G}(Δ)^{3}Z_{G}(−Δ) = 0.)

Let us express E(Σ_{2}^{γ})^{2} as the linear combination of the quantities Σ_{3}^{γ}, and let us bound each term Σ_{3}^{γ} in the above way. This supplies an upper bound for E(Σ_{2}^{γ})^{2} by means of a sum of terms of the form (5.11). 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(Σ_{2}^{γ})^{2} ≤ K_{1}^{2}K_{2}^{2}C(n,m)Cε ∑_{r=1}^{n+m} ∑_{j_{1},...,j_{r}}′′′G(Δ_{j_{1}})··· G(Δ_{j_{r}}),

where the indices j_{1},...,j_{r} ∈ {±1,...,±N} in the sum ∑′′′are all different, and K_{j} = sup|h_{j}(x)|, j = 1,2. Hence

E(Σ_{2}^{γ})^{2} ≤ C_{1}ε ∑_{r=1}^{n+m}G(A)^{r} ≤ C_{2}ε

with some appropriate constants C_{1} and C_{2}. Because of the inequality (5.9), the identity n!I_{G}(h_{1})m!I_{G}(h_{2}) = ∑_{γ∈Γ} (Σ_{1}^{γ}+ Σ_{γ}^{2}) and the last relation one has

E ( n!I_{G}(h_{1})m!I_{G}(h_{2}) − ∑_{γ∈Γ} (n + m − 2|γ|)!I_{G}(h_{γ})) ^{2}

= E ( ∑_{γ∈Γ} (Σ_{1}^{γ}+ Σ_{2}^{γ}− (n + m − 2|γ|)!I_{G}(h_{γ})) ) ^{2}

≤ C_{3}( ∑_{γ∈Γ} E((m + n − 2|γ|)!I_{G}(h_{γ}) − Σ_{1}^{γ})^{2}+ E(Σ_{2}^{γ})^{2}) ≤ C_{4}ε.

Since ε > 0 can be chosen arbitrarily small, Part B is proved in the special case h_{1} ∈ ˆ ¯H_{G}^{n}, h_{2} ∈ ˆ ¯H_{G}^{m}.

If h_{1} ∈ H̄_{G}^{n} and h_{2} ∈ H̄_{G}^{m}, then let us choose a sequence of functions h_{1,r} ∈ ˆ ¯H_{G}^{n} and h_{2,r} ∈ ˆ ¯H_{G}^{m} such that h_{1,r} → h_{1} and h_{2,r} → h_{2} in the norm of the spaces H̄_{G}^{n} and H̄_{G}^{m} respectively. Define the functions ĥ_{γ}(r) and h_{γ}(r) in the same way as h_{γ}, 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 by 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})| → 0,

and

E|I_{G}(h_{γ}) − I_{G}(h_{γ}(r))| → 0 for all γ ∈ Γ

as r → ∞. Then a simple limiting procedure shows that Theorem 5.3 holds for all h_{1} ∈ H̄_{G}^{n} and h_{2} ∈ H̄_{G}^{m}.

We have

E|I_{G}(h_{1})I_{G}(h_{2}) − I_{G}(h_{1,r})I_{G}(h_{2,r})|

≤ 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})|

≤ \frac{1}{n!m!}( ∥h_{1}− h_{1,r}∥^{1/2}∥h_{2}∥^{1/2}+ ∥h_{2}− h_{2,r}∥^{1/2}∥h_{1,r}∥) → 0,

and by Part (A) of Theorem 5.3

E|I_{G}(h_{γ}) − I_{G}(h_{γ}(r))| ≤ E|I_{G}(h_{γ}) − I_{G}(ĥ_{γ}(r))| + E|I_{G}(h_{γ}(r)) − I_{G}(ĥ_{γ}(r))|
≤ ∥h_{γ}− ĥ_{γ}(r)∥^{1/2}+ ∥h_{γ}(r) − ĥ_{γ}(r)∥^{1/2}
≤ ∥h_{1}− ĥ_{1,r}∥^{1/2}∥h_{2}∥^{1/2}+ ∥h_{2}− ĥ_{2,r}∥^{1/2}∥h_{1,r}∥^{1/2} → 0.

Theorem 5.3 is proved. ⊓ ⊔

We formulate some consequences of Theorem 5.3. Let Γ̄ ⊂ Γ denote the set of complete diagrams, i.e. let a diagram γ ∈ Γ̄ if an edge enters in each vertex of γ. We have EI(h_{γ}) = 0 for all γ ∈ Γ \ Γ̄, since (4.3) holds for all f ∈ H̄_{G}^{n}, n ≥ 1. If γ ∈ Γ̄, then I(h_{γ}) ∈ H̄_{G}^{0} . Let h_{γ} denote the value of I(h_{γ}) in this case. Now we have the following

Corollary 5.4. For all h_{1} ∈ H̄_{G}^{n_{1}} ,…, h_{n} ∈ H̄_{G}^{n_{m}}

En_{1}!I_{G}(h_{1})··· n_{m}!I_{G}(h_{m}) = ∑_{γ∈Γ̄}h_{γ}.

(The sum on the right-hand side equals zero if Γ̄ is empty.)

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 ≤ k ≤ p, 1 ≤ j ≤ n_{k}, be a sequence of Gaussian random variables. We want to calculate the expected value of the Wick polynomials : X_{k,1}··· X_{k,n_{k}}: , 1 ≤ k ≤ p, if we know all covariances EX_{k,j}X_{k̄,j̄}= a((k,j),(k̄,j̄)), 1 ≤ k,k̄,≤ p, 1 ≤ j ≤ n_{k}, 1 ≤ j̄≤ n̄_{k}. For this goal let us consider the class of closed diagrams Γ̄(k_{1},...,k_{p}), and define the following quantity γ(A) depending on the closed diagrams γ and the set A of all covariances EX_{k,j}X_{k̄,j̄}= a((k,j),(k̄,j̄))

γ(A) = ∏_{((k,j),(k̄,j̄)) is an edge of γ}a((k,j),(k̄,j̄)), γ ∈ Γ .

With the above notation we can formulate the following result.

Corollary 5.5. Let X_{k,j}, EX_{k,j} = 0, 1 ≤ k ≤ p, 1 ≤ j ≤ n_{k}, be a sequence of Gaussian random variables. Let a((k,j),(k̄,j̄)) = EX_{k,j}X_{k̄,j̄}, 1 ≤ k,k̄,≤ p, 1 ≤ j ≤ n_{k}, 1 ≤ j̄≤ n̄_{k}denote the covariances of these random variables. Then the expected value of the product of the Wick polynomials : X_{k,1}··· X_{k,n_{k}}: , 1 ≤ k ≤ p, can be expressed as

E ( ∏_{k=1}^{p} : X_{k,1}··· X_{k,n_{k}}: ) = ∑_{γ∈Γ̄(k_{1},...,k_{p})}γ(A)

with the above defined quantities γ(A). In the case when Γ̄(k_{1},...,k_{p}) is empty, e.g. if k_{1}+ ··· + k_{p} is an odd number, the above expectation equals zero.

Remark. In the special case when X_{k,1} = ···= X_{k,n_{k}} = X_{k}, and EX_{k}^{2} = 1 for all indices 1 ≤ k ≤ 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),(k̄,j̄)) = ā(k,k̄) with a function ā(·,·) not depending on the arguments j and j̄, and the left-hand side of the identity in Corollary 5.5 equals EH_{n_{1}}(X_{1})··· H_{n_{p}}(X_{p}) with standard normal random variables X_{1},...,X_{n} with correlations EX_{k}Xk̄ = ā(k,k̄).

Proof of Corollary 5.5. We can represent the random variables X_{k,j} in the form X_{k,j} = ∑_{p} c_{k,j,p}ξ_{p} with some appropriate coefficients c_{k,j,p}, where ξ_{1},ξ_{2},... is a se-

quence 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) = \frac{1}{2π} for |x| < π, and g(x) = 0 for |x| ≥ π. The random integrals ∫ e^{ipx}Z(dx), p = 0,±1,±2,..., are independent standard normal random variables. Define h_{k,j}(x) = ∑_{p} c_{k,j,p}e^{ipx}, k = 1,...,p, 1 ≤ j ≤ n_{k}. The random vari-

ables X_{k,j} can be identified with the random integrals ∫ h_{k,j}(x)Z(dx), k = 1,...,p,

1 ≤ j ≤ n_{k}, since their joint distributions coincide. Put ĥ_{k}(x_{1},...,x_{n_{k}}) = ∏_{j=1}^{n_{k}} h_{k,j}(x_{j}).

It follows from Theorem 4.7 that

: X_{k,1}··· X_{k,n_{k}}: = ∫ ĥ_{k}(x_{1},...,x_{n_{k}})Z(dx_{1})...Z(dx_{n_{k}}) = n_{k}!I(ĥ_{k}(x_{1},...,x_{n_{k}}))

for all 1 ≤ k ≤ p. Hence an application of Corollary 5.4 yields Corollary 5.5. One only has to observe that ∫_{−π}^{π} h_{k,j}(x)\overline{h_{k̄,j̄}(x)}dx = a((k,j),(k̄,j̄)) for all k, k = 1,...,p and 1 ≤ j ≤ n_{k}. ⊓ ⊔

Theorem 5.3 states in particular that the product of Wiener–Itô integrals with respect to a random spectral measure of a stationary Gaussian fields belongs to the Hilbert space H defined by this field, since it can be written as a sum of Wiener– Itô 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} ∈ H_{G}^{n_{1}},…, h_{m} ∈ H_{G}^{n_{m}}. Let |Γ̄(n_{1},n_{1},...,n_{m},n_{m})| denote the number of complete diagrams in Γ̄(n_{1},n_{1},...,n_{m},n_{m}), and put

C(n_{1},...,n_{m}) = \frac{|Γ̄(n_{1},n_{1},...,n_{m},n_{m})|}{n_{1}!··· n_{m}!}.

In the special case n_{1} = ···= n_{m}= n let C̄(n,m) = C(n_{1},...,n_{m}). Then

Corollary 5.6.

E [(n_{1}!I_{G}(h_{1}))^{2}··· (n_{m}!I_{G}(h_{m}))^{2}]
≤ C(n_{1},...,n_{m})E(n_{1}!I_{G}(h_{1}))^{2}··· (n_{m}!E(I_{G}(h_{m}))^{2}.

In particular,

E [(n!I_{G}(h))^{2m}]≤ C̄(n,m)(E(n!I_{G}(h))^{2})^{m} if h ∈ H_{G}^{n}.

Corollary 5.6 follows immediately from Corollary 5.4 by applying it first for the sequence h_{1},h_{1},...,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}, 1 ≤ j ≤ m.

One only has to observe that |h_{γ}| ≤ ∥h_{1}∥^{2}··· ∥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} ∈ H_{G}^{n_{1}},…, h_{m} ∈ H_{G}^{n_{m}} are constant functions, then equality can be written in Corollary 5.6. We remark that in this case I_{G}(h_{1}),...,I_{G}(h_{m}) are constant times the n_{1}-th,…, n_{m}-th Hermite polynomials of the same standard normal random variable. Let us emphasize that the constant C(n_{1},...,n_{m}) depends only on the parameters n_{1},...,n_{m} and not on the form of the functions h_{1},...,h_{m}. The function C(n_{1},...,n_{m}) is monotone in its arguments. The following argument shows that

C(n_{1}+ 1,n_{2},...,n_{m})≥ C(n_{1},...,n_{m})

Let us call two complete diagrams in Γ̄(n_{1},n_{1},...,n_{m},n_{m}) or in Γ̄(n_{1} + 1,n_{1} + 1,...,n_{m},n_{m}) equivalent if they can be transformed into each other by permuting the vertices (1,1),...,(1,n_{1}) in Γ̄(n_{1},n_{1},...,n_{m},n_{m}) or the vertices (1,1),...,(1,n_{1} + 1) in Γ̄(n_{1} + 1,n_{1} + 1,...,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

\frac{1}{n_{1}!}|Γ̄(n_{1},n_{1},...,n_{m},n_{m})| ≤ \frac{1}{(n_{1}+ 1)!} |Γ̄(n_{1}+ 1,n_{1}+ 1,...,n_{m},n_{m})|

as we claimed.

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

Corollary 5.7. Let ξ_{1},...,ξ_{k} be a normal random vector, and P(x_{1},...,x_{k}) a polynomial of degree n. Then

E [P(ξ_{1},...,ξ_{k})^{2m}]≤ C̄(n,m)(n + 1)^{m}(EP(ξ_{1},...,ξ_{k})^{2})^{m}

with the constant C̄(n,m) introduced before Corollary 5.6.

The multiplying constant C̄(n,m)(n + 1)^{m} is not sharp in this case.

Proof of Corollary 5.7. We can write ξ_{j} = ∫ f_{j}(x)Z(dx) with some f_{j} ∈ H ^{1}, j = 1,2,...,k, where Z(dx) is the same as in the proof of Corollary 5.5. There exist some h_{j} ∈ H ^{j}, j = 0,1,...,n, such that

P(ξ_{1},...,ξ_{k}) = ∑_{j=0}^{n} j!I(h_{j}).

5 The proof of Itô’s formula. The diagram formula and some of its consequences 59

Then

EP(ξ_{1},...,ξ_{k})^{2m}= E [ ( ∑_{j=0}^{n} j!I(h_{j})) ^{2m}] ≤ (n + 1)^{m}E [ ∑_{j=0}^{n} (j!I(h_{j}))^{2}]^{m}

≤ (n + 1)^{m} ∑_{p_{1}+··· +p_{n}=m} C(p_{1},...,p_{n})(EI(h_{0})^{2})^{p_{0}} ··· (En!I(h_{n})^{2})^{p_{n}} \frac{m!}{p_{1}!··· p_{n}!}

≤ (n + 1)^{m}C̄(n,m) ∑_{p_{1}+··· +p_{n}=m} (EI(h_{0})^{2})^{p_{0}} ··· (EI(n!h_{n})^{2})^{p_{n}} \frac{m!}{p_{1}!··· p_{n}!}

= (n + 1)^{m}C̄(n,m)[∑ E(j!I(h_{j}))^{2}]^{m}= (n + 1)^{m}C̄(n,m)(EP(ξ_{1},...,ξ_{k})^{2})^{m}.

⊓ ⊔

Subordinated random fields. Construction of self-similar fields

Let X_{n}, n ∈ ℤ_{ν}, be a discrete stationary Gaussian random field, and let the random field ξ_{n}, n ∈ ℤ_{ν}, 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 ξ_{0} can be represented as

ξ_{0} = f_{0}+ ∑_{k=1}^{∞} \frac{1}{k!} ∫ f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k})

with an appropriate f = (f_{0},f_{1},...) ∈ ExpH_{G} in a unique way. This formula together with Theorem 4.4 yields the following

Theorem 6.1. A random field ξ_{n}, n ∈ ℤ_{ν}, subordinated to the stationary Gaussian random field X_{n}, n ∈ ℤ_{ν}, can be written in the form

ξ_{n}= f_{0}+ ∑_{k=1}^{∞} \frac{1}{k!} ∫ e^{i((n,x_{1}+··· +x_{k})} f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}), n ∈ Z_{ν}, (6.1)

with some f = (f_{0},f_{1},...) ∈ ExpH_{G}, where G is the spectral measure of the field X_{n}, and Z_{G} is the random spectral measure adapted to it. This representation is unique. It is also clear that formula (6.1) defines a subordinated field for all f ∈ ExpH_{G}.

If the spectral measure G has the property G({x: x_{p} = u}) = 0 for all u ∈ R^{1} and 1 ≤ p ≤ ν, where x = (x_{1},...,x_{ν}) (this is a strengthened form of the non-atomic property), then the functions

f̄_{k}(x_{1},...,x_{k}) = f_{k}(x_{1},...,x_{k})χ̃_{0}^{−1} (x_{1}+ ··· + x_{k}), k = 1,2,...,

are meaningful, as functions in the measure space (R^{kν},B^{kν},G^{k}), where χ̃_{n}(x) =

e^{i(n,x)} ∏_{p=0}^{ν} \frac{e^{ix^{(p)}}−1}{ix^{(p)}} , n ∈ ℤ_{ν}, denotes the Fourier transform of the indicator function

of the ν-dimensional unit cube ∏_{p=1}^{ν} [n^{(p)},n^{(p)} + 1]. Then the random variable ξ_{n} in formula (6.1) can be rewritten in the form

ξ_{n} = f_{0}+ ∑_{k=1}^{∞} \frac{1}{k!} ∫ χ̃_{n}(x_{1}+ ··· + x_{k})f̄_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}).

Hence the following Theorem 6.1′can be considered as the continuous time version of Theorem 6.1.

Theorem 6.1′. Let a generalized random field ξ(φ), φ ∈ S, be subordinated to a stationary Gaussian generalized random field X(φ), φ ∈ S. Let G denote the spectral measure of the field X(φ), and let Z_{G} be the random spectral measure adapted to it. Then ξ(φ) can be written in the form

ξ(φ) = f_{0}· φ̃(0) + ∑_{k=1}^{∞} \frac{1}{k!} ∫ φ̃(x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}),

(6.2)

where the functions f_{k} are invariant under all permutations of their variables,

f_{k}(−x_{1},...,−x_{k}) = \overline{f_{k}(x_{1},...,x_{k})}, k = 1,2,...,

and

∑_{k=1}^{∞} \frac{1}{k!} ∫ (1 + |x_{1}+ ··· + x_{k}|^{2})^{−p} |f_{k}(x_{1}+ ··· + x_{k})|^{2}G(dx_{1})...G(dx_{k}) < ∞  (6.3)

with an appropriate number p > 0. This representation is unique.

Contrariwise, all random fields ξ(φ), φ ∈ S, defined by formulas (6.2) and (6.3) are subordinated to the stationary, Gaussian random field X(φ), φ ∈ S.

Proof of Theorem 6.1′. The proof 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 [15].) In particular, we exploit the following continuity property of generalized random fields and subordinated generalized random fields. If φ_{n} → φ in the topology of the Schwartz space S, and X(φ), φ ∈ S, is a generalized random field, then X(φ_{n}) ⇒ X(φ) stochastically. If X(φ), φ ∈ S, is a generalized Gaussian random field, then also the relation E[X(φ_{n}) − X(φ)]^{2} → 0 holds in this case. Similarly, if ξ(φ), φ ∈ S, is a subordinated generalized random field, and φ_{n} → φ, then E[ξ(φ_{n}) − ξ(φ)]^{2} → by the definition of subordinated fields.

It can be seen with some work that a random field ξ(φ), φ ∈ S, defined by (6.2) and (6.3) is subordinated to X(φ). One has to check that the definition of ξ(φ) in formula (6.2) is meaningful for all φ ∈ S, because of (6.3), ξ(T_{t}φ) = T_{t}ξ(φ) for all shifts T_{t}, t ∈ R^{ν}, by Theorem 4.4, and also the following continuity property holds. For all ε > 0 there is a small neighbourhood H of the origin in the space S such that if φ = φ_{1}− φ_{2} ∈ H for some φ_{1},φ_{2} ∈ S then E[ξ(φ_{1}) − ξ(φ_{2})]^{2} = Eξ(φ)^{2} < ε^{2}.

Since the Fourier transform φ(·) → φ̃(·) is a bicontinuous map in S, to prove the above continuity property it is enough to check that Eξ(φ)^{2} < ε^{2} if φ̃ ∈ H for an appropriate small neighbourhood H of the origin in S. But this relation holds with the choice H = \{φ: (1 + |x|^{2})^{p}|φ(x)| ≤ \frac{ε^{2}}{K} for all x ∈ R^{ν}\} with a sufficiently large K > 0 because of condition (6.3).

To prove that all subordinated fields have the above representation observe that the relation

ξ(φ) = Ψ_{φ,0} + ∑_{k=1}^{∞} \frac{1}{k!} ∫ Ψ_{φ,k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k})  (6.4)

holds for all φ ∈ S with some (Ψ_{φ,0},Ψ_{φ,1},...) ∈ ExpH_{G} depending on the function φ. We are going to show that these functions Ψ_{φ,k} can be given in the form

Ψ_{φ,k}(x_{1},...,x_{k}) = f_{k}(x_{1},...,x_{k}) · φ̃(x_{1}+ ··· + x_{k}), k = 1,2,...,

with some functions f_{k} ∈ B^{kν}, and

Ψ_{φ,0} = f_{0}· φ̃(0)

for all φ ∈ S with a sequence of functions f_{0},f_{1},... not depending on φ.

To show this let us choose a φ_{0} ∈ S such that φ̃_{0}(x) > 0 for all x ∈ R^{ν}. (We can make for instance the choice φ_{0}(x) = e^{−(x,x)}.) We claim that the finite linear combinations ∑ a_{p}φ_{0}(x − t_{p}) = ∑ a_{p}T_{t_{p}}φ_{0}(x) are dense in S. To prove this it is enough to show that the functions ψ whose Fourier transforms ψ̃ have a compact support can well be approximated by such linear combinations, because these functions ψ are dense in S. (The statement that these functions ψ are dense in S is equivalent to the statement that their Fourier transform ψ̃ are dense in the space S̃⊂ S^{c} consisting of the Fourier transforms of the (real valued) functions in the space S.) We have \frac{ψ̃}{φ̃_{0}} ∈ S^{c} for such functions ψ, where S^{c} denotes the Schwartz-space of complex valued, at infinity strongly decreasing, smooth functions again, because φ̃_{0}(x) ≠ 0, and ψ̃ has a compact support. There exists a function χ ∈ S such that χ̃ = \frac{ψ̃}{φ̃_{0}} . (Here we exploit that the space of Fourier transforms of the functions from

S agrees with the space of those functions f ∈ S^{c} for which f(−x) = \overline{f(x)}.) Therefore ψ(x) = χ ∗ φ_{0}(x) = ∫ χ(t)φ_{0}(x − t)dt, where ∗ denotes convolution. It can be seen by exploiting this relation together with the rapid decrease of χ and φ_{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 ε > 0 there exists a finite linear combination ψ̂(x) = ψ̂_{r,s,ε} (x) = ∑_{p} a_{p}φ_{0}(x − t_{p}) such that

(1+|x|^{s})|ψ(x)−ψ̂(x)| < ε for all x ∈ R^{ν}, and the same estimate holds for all derivatives of ψ(x) − ψ̂(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 ψ = χ ∗φ_{0} with a function χ ∈ D, which has compact support. To give the desired approximation choose a small number δ > 0, introduce the cube Δ = Δ(δ) = [−δ,δ)^{ν} ⊂ R^{ν} and define the vectors k(δ) = (2k_{1}δ,...,2k_{ν}δ) ∈ R^{ν} for all k = (k_{1},...,k_{ν}) ∈ ℤ_{ν}. Given a fixed vector x ∈ R^{ν} let us define the vector u(x) ∈ R^{ν} for all u ∈ R^{ν} as u(x) = x + k(δ) with that vector k ∈ ℤ_{ν} for which x + k(δ) − u ∈ Δ, and put φ_{0,x}(u) = φ_{0}(u(x)). It can be seen that ψ̂(x) = χ ∗ φ_{0,x}(x) is a finite linear combination of numbers of the form φ_{0}(x − t_{k}) (with t_{k} = k(δ)) with coefficients not depending on x. Moreover, if δ > 0 is chosen sufficiently small (depending on r,s and ε), then ψ̂(x) = ψ̂_{r,s,ε} (x) has all properties we demanded.

The above argument implies that there is a sequence of functions ψ̂_{r,s,ε} which converges to the function ψ in the topology of the space S. As a consequence, the finite linear combinations ∑ a_{p}φ_{0}(x − t_{p}) are dense in S.

Define

f_{k}(x_{1},...,x_{k}) = \frac{Ψ_{φ_{0},k}(x_{1},...,x_{k})}{φ̃_{0}(x_{1}+ ··· + x_{k})}, k = 1,2,..., and f_{0} = \frac{Ψ_{φ_{0},0}}{φ̃_{0}(0)}.

If φ(x) = ∑ a_{p}φ_{0}(x − t_{p}) = ∑ a_{p}T_{t_{p}}φ_{0}(x), and the sum defining φ is finite, then by Theorem 4.4

ξ(φ) = (∑ a_{p})f_{0}· φ̃_{0}(0) + ∑_{k=1}^{∞} \frac{1}{k!} ∫ ∑_{p} a_{p}e^{i(t_{p},x_{1}+··· +x_{k})} φ̃_{0}(x_{1}+ ··· + x_{k})
·f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k})

= f_{0}· φ̃(0) + ∑_{k=1}^{∞} \frac{1}{k!} ∫ φ̃(x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}).

Relation (6.4) holds for all φ ∈ S, and there exists a sequence φ_{j}(x) = ∑_{p} a_{p}^{(j)}φ_{0}(x−

t_{p}^{(j)}) ∈ S satisfying (6.2) such that φ_{j} → φ in the topology of S. This implies that limE[ξ(φ_{j}) − ξ(φ)]^{2} → 0, and in particular EI_{G}(Ψ_{φ,k} − φ̂_{j,k}f_{k})^{2} → 0 with φ̂_{j,k}(x_{1},...,x_{k}) = φ̃_{j} (x_{1}+ ··· + x_{k}) as j → ∞ for all k = 1,2,.... (To carry out some further argument we restrict the domain of integration to a bounded set A.) We get that

∫_{A}|Ψ_{φ,k}(x_{1},...,x_{k}) − φ̃_{j} (x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k})|^{2}G(dx_{1})...G(dx_{k}) → 0

as j → ∞ for all k and for all bounded sets A ∈ R^{kν}. On the other hand,

∫_{A}|φ̃(x_{1}+ ··· + x_{k}) − φ̃_{j} (x_{1}+ ··· + x_{k})|^{2}|f_{k}(x_{1},···,x_{k})|^{2}G(dx_{1})...G(dx_{k}) → 0,

since φ̃_{j} (x)−φ̃(x) → 0 in the supremum norm if φ̃_{j} → φ̃ in the topology of S, and the property φ̃_{0}(x) > 0 (of the function φ̃_{0} appearing in the definition of the function f_{k}) together with the continuity of φ̃_{0} and the inequality EI_{G}(φ̂_{0,k}f_{k})^{2} < ∞ imply that ∫_{A}|f_{k}(x_{1},...,x_{k})|^{2}G(dx_{1})...G(dx_{k}) < ∞ on all bounded sets A. The last two relations yield that

Ψ_{φ,k}(x_{1},...,x_{k}) = φ̃(x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k}), k = 1,2,...,

since both sides of this identity is the limit of the sequence

φ̃_{j} (x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k}), j = 1,2,...

in the L_{G_{A}^{k}}^{2} norm, where G_{A}^{k}denotes the restriction of the measure G^{k} to the set A.

Similarly,

ψ_{φ,0} = φ̃(0)f_{0}.

These relations imply (6.2).

To complete the proof of Theorem 6.1′ we show that (6.3) follows from the continuity of the transformation F : φ → ξ(φ) from the space S into the space L^{2}(Ω,A ,P).

We recall that the transformation φ → φ̃ is bicontinuous in S^{c}. Hence for a subordinated field ξ(φ), φ ∈ S, the transformation φ̃ → ξ(φ) is a continuous map from the space of the Fourier transforms of the functions in the space S to L^{2}(Ω,A ,P). This continuity implies that there exist some integers p > 0, r > 0 and real number δ > 0 such that if

(1 + |x^{2}|)^{p} | \frac{∂^{s_{1}+··· +s_{ν}}}{∂x^{(1)}^{s_{1}} ...∂x^{(ν)}^{s_{ν}}} φ̃(x)|< δ for all s_{1}+ ··· + s_{ν} ≤ r,  (6.5)

then Eξ(φ)^{2} ≤ 1.

Let us choose a function ψ ∈ S such that ψ has a compact support, ψ(x) = ψ(−x), ψ(x) ≥ 0 for all x ∈ R^{ν}, and ψ(x) = 1 if |x| ≤ 1. (There exist such functions.) Define the functions φ̃_{m}(x) = C(1 + |x|^{2})^{−p} ψ(\frac{x}{m}). Then φ_{m} ∈ S, since its Fourier transform φ̃_{m} is an even function, and it is in the space S being an infinite many times differentiable function with compact support. Moreover, φ_{m} satisfies (6.5) for all m = 1,2,... 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

Eξ(φ_{m})^{2} = ∑ \frac{1}{k!} ∫ |φ̃_{m}(x_{1}+ ··· + x_{k})|^{2}|f_{k}(x_{1},···,x_{k})|^{2}G(dx_{1})...G(dx_{k}) ≤ 1

for all m = 1,2,....

As φ̃_{m}(x) → C(|1 + |x|^{2})^{−p} as m → ∞, and φ̃_{k}(x) ≥ 0, an m → ∞ limiting procedure in the last relation together with Fatou’s lemma imply that

C ∑ \frac{1}{k!} ∫ (1 + |x_{1}+ ··· + x_{k})|^{2})^{−p} |f_{k}(x_{1},···,x_{k})|^{2}G(dx_{1})...G(dx_{k}) ≤ 1.

Theorem 6.1′is proved. ⊓ ⊔

We shall call the representations given in Theorems 6.1 and 6.1′ the canonical representation of a subordinated field. From now on we restrict ourselves to the case Eξ_{n} = 0 or Eξ(φ) = 0 respectively, i.e. to the case when f_{0} = 0 in the canonical representation. If

ξ(φ) = ∑_{k=1}^{∞} \frac{1}{k!} ∫ φ̃(x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}),

then

ξ(φ_{t}^{A}) = ∑_{k=1}^{∞} \frac{1}{k!} \frac{t^{ν}}{A(t)}∫ φ̃(t(x_{1}+ ··· + x_{k}))f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k})

with the function φ_{t}^{A}defined in (1.3). Define the spectral measures G_{t}by the formula G_{t}(A) = G(tA). Then we have by Lemma 4.6

ξ(φ_{t}^{A}) =^{Δ} ∑_{k=1}^{∞} \frac{1}{k!} \frac{t^{ν}}{A(t)}∫ φ̃(x_{1}+ ··· + x_{k})f_{k}( \frac{x_{1}}{t},..., \frac{x_{k}}{t}) Z_{G_{t}}(dx_{1})...Z_{G_{t}}(dx_{k}).

If G(tB) = t^{2κ}G(B) with some κ > 0 for all t > 0 and B ∈ B^{ν}, f_{k}(λx_{1},...,λx_{k}) = λ^{ν−κk−α} f_{k}(x_{1},...,x_{k}), and A(t) is chosen as A(t) = t^{α}, then Theorem 4.5 (with the choice G′(B) = G(tB) = t^{2κ}G(B)) implies that ξ(φ_{t}^{A}) =^{Δ} ξ(φ). Hence we obtain the following

Theorem 6.2. Let a generalized random field ξ(φ) be given by the formula

ξ(φ) = ∑_{k=1}^{∞} \frac{1}{k!} ∫ φ̃(x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}).  (6.6)

If f_{k}(λx_{1},...,λx_{k}) = λ^{ν−κk−α} f_{k}(x_{1},...,x_{k}) for all k, (x_{1},...,x_{k}) ∈ R^{kν} and λ > 0, G(λA) = λ^{2κ}G(A) for all λ > 0 and A ∈ B^{ν}, then ξ is a self-similar random field with parameter α.

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

Theorem 6.2′. If a discrete random field ξ_{n}, n ∈ ℤ_{ν}, has the form

ξ_{n} = ∑_{k=1}^{∞} \frac{1}{k!} ∫ χ̃_{n}(x_{1}+ ··· + x_{k})f_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}), n ∈ ℤ_{ν}, (6.7)

and f_{k}(λx_{1},...,λx_{k}) = λ^{ν−κk−α} f_{k}(x_{1},...,x_{k}) for all k, G(λA) = λ^{2κ}G(A), then ξ_{n} is a self-similar random field with parameter α.

Theorems 6.2 and 6.2′enable us to construct self-similar random fields. Nevertheless, we have to check whether formulas (6.6) and (6.7) are meaningful. The hard part of this problem is to check whether

∑ \frac{1}{k!} ∫ |χ̃_{n}(x_{1}+ ··· + x_{k})|^{2}|f_{k}(x_{1},...,x_{k})|^{2}G(dx_{1})...G(dx_{k}) < ∞,

or whether

∑ \frac{1}{k!} ∫ |φ̃(x_{1}+ ··· + x_{k})|^{2}|f_{k}(x_{1},...,x_{k})|^{2}G(dx_{1})...G(dx_{k}) < ∞ for all φ ∈ S .

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

Let us define the measure G

G(A) = ∫_{A}|x|^{2κ−ν} a( \frac{x}{|x|}) dx, A ∈ B^{ν},  (6.8)

where a(·) is a non-negative, measurable and even function on the ν-dimensional unit sphere S_{ν−1} , and κ > 0. (The condition κ > 0 is imposed to guarantee the relation G(A) < ∞ for all bounded sets A ∈ B^{ν}.) We prove the following

Proposition 6.3. Let the measure G be the same as in formula (6.8).

(a) If the function a(·) is bounded on the unit sphere S_{ν−1} , and \frac{ν}{k} > 2κ > 0, then

D(n) = ∫ |χ̃_{n}(x_{1}+ ··· + x_{k})|^{2}G(dx_{1})...G(dx_{k}) < ∞ for all n ∈ ℤ_{ν},

and

D(φ) = ∫ |φ̃(x_{1}+ ··· + x_{k})|^{2}G(dx_{1})...G(dx_{k})
≤ C ∫ (1 + |x_{1}+ ··· + x_{k})|^{2})^{−p} G(dx_{1})...G(dx_{k}) < ∞

for all φ ∈ S and p > \frac{ν}{2} with some C = C(φ,p) < ∞.

(b) If there is a constant C > 0 such that a(x) > C in a neighbourhood of a point x_{0} ∈ S_{ν−1} , and either 2κ ≤ 0 or 2κ ≥ \frac{ν}{k}, then the integrals D(n) are divergent, and the same relation holds for D(φ) with some φ ∈ S.

Proof of Proposition 6.3. Proof of Part (a) We may assume that a(x) = 1 for all x ∈ S_{ν−1} . Define

J_{κ,k}(x) = ∫_{x_{1}+··· +x_{k}=x} |x_{1}|^{2κ−ν} ··· |x_{k}|^{2κ−ν} dx_{1}... dx_{k}, x ∈ R^{ν},

for k ≥ 2, where dx_{1}... dx_{k} denotes the Lebesgue measure on the hyperplane x_{1}+ ··· + x_{k} = x, and let J_{κ,1}(x) = |x|^{2κ−ν} . We have

J_{κ,k}(λx) = |λ|^{k(2κ−ν)+(k−1)ν} J_{κ,k}(x),= |λ|^{2kκ−ν} J_{κ,k}(x), x ∈ R^{ν} λ > 0,

because of the homogeneity of the integral. We can write, because of (6.8) with a(x) ≡ 1

D(n) = ∫_{R^{ν}}|χ̃_{n}(x)|^{2}J_{κ,k}(x)dx,  (6.9)

and

D(φ) = ∫_{R^{ν}}|φ̃(x)|^{2}J_{κ,k}(x)dx.

We prove by induction on k that

J_{κ,k}(x) ≤ C(κ,k)|x|^{2κk−ν}  (6.10)

with an appropriate constant C(κ,k) < ∞ if \frac{ν}{k} > 2κ > 0.

We have

J_{κ,k}(x) = ∫ J_{κ,k−1} (y)|x − y|^{2κ−ν} dy.

Hence

J_{κ,k} ≤ C(κ,k − 1)∫ |y|^{(2κ(k−1)−ν} |x − y|^{2κ−ν} dy
= C(κ,k − 1)|x|^{2κk−ν} ∫ |y|^{(2κ(k−1)−ν} | \frac{x}{|x|}− y|^{2κ−ν} dy = C(κ,k)|x|^{2κk−ν} ,

since ∫ |y|^{(2κ(k−1)−ν} | \frac{x}{|x|}− y|^{2κ−ν} dy < ∞.

The last integral is finite, since its integrand behaves at zero asymptotically as C|y|^{2κ(k−1)−ν} , at the point e = \frac{x}{|x|} ∈ S_{ν−1} as C_{2}|y − e|^{2κ−ν} and at infinity as C_{3}|y|^{2κk−2ν} . Relations (6.9) and (6.10) imply that

D(n)≤ C′∫ |χ̃_{0}(x)|^{2}|x|^{2κk−ν} dx ≤ C′′∫ |x|^{2κk−ν} ∏_{l=1}^{ν} \frac{1}{1 + |x^{(l)}|^{2}} dx

≤ C′′′∫_{|x^{(1)}|= max_{1≤l≤ν} |x^{(l}|} |x^{(1)}|^{2κk−ν} ∏_{l=1}^{ν} \frac{1}{1 + |x^{(l)}|^{2}} dx

= ∑_{p=0}^{∞} C′′′∫_{|x^{(1)}|= max_{1≤l≤ν} |x^{(l}|, 2^{p}≤|x^{(1)}|<2^{p+1}} +C′′′∫_{|x^{(1)}|= max_{1≤l≤ν} |x^{(l}|,|x^{(1)}|<1} .

The second term in the last sum can be simply bounded by a constant, since B =

\{ x: |x^{(1)}| = max_{1≤l≤ν} |x^{(l}|, |x^{(1)}| < 1\} ⊂ \{x: |x| ≤ \sqrt{ν}\}, and |x^{(1)}|^{2κk−ν} ∏_{l=1}^{ν} \frac{1}{1+|x^{(l)}|^{2}} ≤

const.|x|^{2κk−ν} on the set B. Hence

D(n) ≤ C_{1} ∑_{p=0}^{∞} 2^{p(2κk−ν)} [∫_{−∞}^{∞} \frac{1}{1 + x^{2}} dx]^{ν} + C_{2} < ∞.

We have |φ(x)| ≤ C(1 + |x^{2}|)^{−p} with some C > 0 and D > 0 if φ ∈ S. The proof of the estimate D(φ) < ∞ for φ ∈ S is similar but simpler.

Proof of Part (b). Define, similarly to the function J_{κ,k} the function

J_{κ,k,a}(x) = ∫_{x_{1}+··· +x_{k}=x} |x_{1}|^{2κ−ν} a( \frac{x_{1}}{|x_{1}|}) ··· |x_{k}|^{2κ−ν} a( \frac{x_{k}}{|x_{k}|}) dx_{1}... dx_{k}, x ∈ R^{ν},

where dx_{1}... dx_{k} denotes the Lebesgue measure on the hyperplane x_{1}+··· +x_{k}= x. Since

J_{κ,k,a}(x) ≥ ∫_{y: |y|<( \frac{1}{2}+α)|x|, |y−x|<( \frac{1}{2}+α)|x|} J_{κ,k−1,a} (y)a( \frac{x − y}{|x − y|} ) |x − y|^{2κ−ν} dy

with an arbitrary α > 0 an argument similar to the one in Part (a) shows that

J_{κ,k,a}(x)\{ \begin{matrix}≥ C̄(κ,k)|x|^{2κk−ν} if \frac{ν}{k} > 2κ > 0, \\ = ∞ if κ ≤ 0 or 2κ ≥ \frac{ν}{k} \end{matrix}

if \frac{x}{|x|} is close to such a point x_{0} ∈ S_{ν−1} in whose small neighbourhood the function a(·) is separated from zero. Since |χ̃_{n}(x)|^{2} > 0 for almost all x ∈ R^{ν},

D(n) = ∫ |χ̃_{n}(x)|^{2}J_{κ,k,a}(x)dx = ∞

under the conditions of Part (b). Similarly D(φ) = ∞ if |φ̃(x)|^{2} > 0 for almost all x ∈ R^{ν}. 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_{ν−1} . ⊓ ⊔

Theorem 6.2 and 6.2′together with Proposition 6.3 have the following

Corollary 6.4. The formulae

ξ_{n} = ∑_{k=1}^{M} ∫ χ̃_{n}(x_{1}+ ··· + x_{k}) ∏_{l=1}^{k} ( |x_{l}|^{−κ+(ν−α)/k} · b_{k}( \frac{x_{l}}{|x_{l}|}) )
Z_{G}(dx_{1})...Z_{G}(dx_{k}), n ∈ ℤ_{ν},

and

ξ(φ) = ∑_{k=1}^{M} ∫ φ̃(x_{1}+ ··· + x_{k}) ∏_{l=1}^{k} ( |x_{l}|^{−κ+(ν−α)/k} · b_{k}( \frac{x_{l}}{|x_{l}|}) )
Z_{G}(dx_{1})...Z_{G}(dx_{k}), φ ∈ S ,

define self-similar random fields with self-similarity parameter α if G is defined by formula (6.8), the parameter α satisfies the inequality \frac{ν}{2} < α < ν, and the functions a(·) (in the definition of the measure G(·) in (6.8), b_{1}(·),… b_{k}(·) are bounded even functions on S_{ν−1} .

The following observation may be useful when we want to prove Corollary 6.4. We can replace ξ_{n} by another random field with the same distribution. Thus we can write, by exploiting Theorem 4.5,

ξ_{n} = ∑_{k=1}^{M} χ̃_{n}(x_{1}+ ··· + x_{k})Z_{G′}(dx_{1})...Z_{G′}(dx_{k}), n ∈ ℤ_{ν},

with random spectral measure Z_{G′}corresponding to the spectral measure G′(dx) = b(\frac{x}{|x|})^{2}|x|^{−2κ+2(ν−α)/k} G(dx) = a(\frac{x}{|x|})b(\frac{x}{|x|})^{2}|x|^{−ν+2(ν−α)/k} dx. In the case of generalized random fields a similar argument can be applied.

Remark 6.5. The estimate on J_{κ,k} and the end of the of Part (a) in Proposition 6.3 show that the self-similar random field

ξ(φ) = ∑_{k=1}^{M} ∫ φ̃(x_{1}+ ··· + x_{k})|x_{1}+ ··· + x_{k}|^{p}u( \frac{x_{1}+ ··· + x_{k}}{|x_{1}+ ··· + x_{k}|} )

∏_{l=1}^{k} ( |x_{l}|^{−κ+(ν−α)/k} · b_{k}( \frac{x_{l}}{|x_{l}|}) )Z_{G}(dx_{1})...Z_{G}(dx_{k}), φ ∈ S ,

and

ξ_{n} = ∑_{k=1}^{M} ∫ χ̃_{n}(x_{1}+ ··· + x_{k})|x_{1}+ ··· + x_{k}|^{p}u( \frac{x_{1}+ ··· + x_{k}}{|x_{1}+ ··· + x_{k}|} )

∏_{l=1}^{k} ( |x_{l}|^{−κ+(ν−α)/k} · b_{k}( \frac{x_{l}}{|x_{l}|}) )Z_{G}(dx_{1})...Z_{G}(dx_{k}), n ∈ ℤ_{ν},

are well defined if G is defined by formula (6.8), a(·), b(·) and u(·) are bounded even functions on S_{ν−1} , \frac{ν}{2} < α < ν, and α − p < ν in the generalized and \frac{ν−1}{2} < α − p < ν is the discrete random field case. The self-similarity parameter of these random fields is α − 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{ν}{2}.

In proving the statement of Remark 6.5 we have to check the integrability conditions needed for the existence of the Wiener–Itô integrals ξ(φ) and ξ_{n}. To check them it is worth remarking that in the proof of Part (a) of Proposition 6.3 we proved the estimate J_{κ̄,k}(x) ≤ C(κ̄,k)|x|^{2κ̄k−ν} . We want to apply this inequality in the present case with the choice κ̄ = \frac{ν−α}{k} . Then arguing similarly to the proof of Part (a) of Proposition 6.3 we get to the problem whether the relations ∫ |χ̃_{n}(x)|^{2}|x|^{2p+2(ν−α)−ν} dx < ∞ and ∫ |φ̃(x)|^{2}|x|^{2p+2(ν−α)−ν} dx < ∞ if φ ∈ 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.

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–Itô integrals 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}|^{−κ+(ν−α)/k} b(\frac{x_{l}}{|x_{l}|}) by b(\frac{x_{l}}{|x_{l}|})g(x_{l})|x_{l}|^{−κ+(ν−α)/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 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 ξ_{n} or ξ(φ) defined in Corollary 6.4. It is clear that all h_{γ} ≥ 0. Moreover, if b_{k}(x) does not vanish for some even number k, then there exists a h_{γ} > 0 in the sum expressing an odd moment of ξ_{n} or ξ(φ). Hence the odd moments of ξ_{n} or ξ(φ) 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–Itô integrals of different order is different.

Theorem 6.6. Let G be a spectral measure and Z_{G} a random spectral measure corresponding to G. For all h ∈ 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_{1}x^{2/m}} ≤ P(|I_{G}(h)| > x) ≤ e^{−K_{2}x^{2/m}}

for all x > x_{0}.

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–Itô 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.

Proof of Theorem 6.6. (a) Proof of the upper estimate.

We have

P(|I_{G}(h)| > x) ≤ x^{2N}E(I_{G}(h)|^{2N}).

By Corollary 5.6

E(I_{G}(h)|^{2N}) ≤ C̄(m,N)[E(I_{G}(h)^{2})]^{N} ≤ C̄(m,N)C_{1}^{N},

and by a simple combinatorial argument we obtain that

C̄(m,N) ≤ \frac{(2Nm − 1)(2Nm − 3)··· 1}{(m!)^{N}},

since the numerator on the right-hand side of this inequality equals the number of complete diagrams |Γ̄(\underbrace{m,...,m}_{2N times})| if vertices from the same row can also be con-

nected. Multiplying the inequalities

(2nM − 2j − 1)(2Nm − 2j − 1 − 2N)··· (2Nm − 2j − 1 − 2N(m − 1)) ≤ (2N)^{m}m!,

j = 1,...,N, we obtain that

C̄(m,N) ≤ (2N)^{mN}.

(This inequality could be sharpened, but it is sufficient for our purpose.) Choose a sufficiently small number α > 0, and define N = [αx^{2/m}], where [·] denotes integer part. With this choice we have

P(|I_{G}(h)| > x) ≤ (x^{−2} (2α)^{m}x^{2})^{N}C_{1}^{N} = [C_{1}(2α)^{m}]^{N} ≤ e^{−K_{2}x^{2/m}},

if α is chosen in such a way that C_{1}(2α)^{m}≤ \frac{1}{e}, K_{2} = \frac{α}{2}, and x > x_{0} with an appropriate x_{0} > 0.

(b) Proof of the lower estimate.

First we reduce this inequality to the following statement. Let Q(x_{1},...,x_{k}) be a homogeneous polynomial of order m (the number k is arbitrary), and ξ = (ξ_{1},...,ξ_{k}) a k-dimensional standard normal variable. Then

P(Q(ξ_{1},...,ξ_{k}) > x) ≥ e^{−Kx^{2/m}}  (6.11)

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

I_{G}(h) = ∑_{j_{1}+··· +j_{l}=m} C_{j_{1},...,j_{l}}^{k_{1},...,k_{l}}H_{j_{1}}(ξ_{k_{1}})··· H_{j_{k}}(ξ_{k_{l}}),  (6.12)

where ξ_{1},ξ_{2},... are independent standard normal random variables, C_{j_{1},...,j_{l}}^{k_{1},...,k_{l}} are appropriate coefficients, and the right-hand side of (6.12) 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 (6.12) under the condition ξ_{k+1} = x_{k+1},ξ_{k+2} = x_{k+2},..., where the numbers x_{k+1},x_{k+2},... are arbitrary. This conditional distribution coincides with the distribution of the random variable Q(ξ_{1},...,ξ_{k},x_{k+1},x_{k+2},...) with probability 1, where the polynomial Q is obtained by substituting ξ_{k+1} = x_{k+1}, ξ_{k+2} = x_{k=2},... into the right-hand side of (6.12). It is clear that all these polynomials Q(ξ_{1},...,ξ_{k},x_{k+1},x_{k+2},...) are of order m if k is sufficiently large. It is sufficient to prove that

P(|Q(ξ_{1},...,ξ_{k},x_{k+1},x_{k+2},...)| > x) ≥ 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(ξ_{1},...,ξ_{k},x_{k+1},x_{k+2},...) = Q_{1}(ξ_{1},...,ξ_{k}) + Q_{2}(ξ_{1},...,ξ_{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–Itô integrals with multiplicity less than m. Hence the already proved part of Theorem 6.6 implies that

P(Q_{2}(ξ_{1},...,ξ_{k}) > x) ≤ e^{−q̄Kx^{2/(m−1)}} .

(We may assume that m ≥ 2). Then an application of relation (6.11) to Q_{1} implies the remaining part of Theorem 6.6, thus it suffices to prove (6.11).

If Q(·) is a polynomial of k variables, then there exist some α > 0 and β > 0 such that

λ ( |Q( \frac{x_{1}}{|x|},..., \frac{x_{k}}{|x|}) | > α) > β,

where |x|^{2} = ∑_{j=1}^{k} x_{j}^{2}, and λ denotes the Lebesgue measure on the k-dimensional unit

sphere S_{k−1}. Exploiting that |ξ| and \frac{ξ}{|ξ|} are independent, \frac{ξ}{|ξ|} is uniformly distributed

on the unit sphere S_{k−1}, and P(|ξ| > x) ≥ ce^{−x^{2}} for a k-dimensional standard normal random variable, we obtain that

P(|Q(ξ_{1},...,ξ_{k})| > x) ≥ βP ( |ξ|^{m}> \frac{x}{α}) > e^{−Kx^{2/m}},

if the constants K and x are sufficiently large. Theorem 6.6 is proved. ⊓ ⊔

Theorem 6.6 implies in particular that Wiener–Itô integrals of different multiplicity have different distributions. A bounded random variable measurable with respect to the σ-algebra generated by a stationary Gaussian field can be expressed as a sum of multiple Wiener–Itô 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–Itô 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 (1.1) or (1.3), when A_{N} = N^{α}, A(t) = t^{α} with some α.) 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–Itô integrals.

On the original Wiener–Itô integral

In this chapter the definition of the original Wiener–Itô integral introduced by Itô in [18] 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,M ,μ) with a σ-finite measure μ be given. Let μ satisfy the following continuity property: For all ε > 0 and A ∈ M, μ(A) < ∞, there exist some disjoint sets B_{j} ∈ M, j = 1,...,N, with some integer N such that μ(B_{j}) < ε

for all 1 ≤ j ≤ N, and A = ⋃_{j=1}^{N} B_{j}. We introduce the following definition.

Definition of (Gaussian) random orthogonal measures. A system of random variables Z_{μ}(A), A ∈ M, μ(A) < ∞, is called a Gaussian random orthogonal measure corresponding to the measure μ if

(i) Z_{μ}(A_{1}),...,Z_{μ}(A_{k}) are independent Gaussian random variables if the sets A_{j} ∈ M, μ(A_{j}) < ∞, j = 1,...,k, are disjoint.

(ii) EZ_{μ}(A) = 0, EZ_{μ}(A)^{2} = μ(A).

(iii) Z_{μ} ( ⋃_{j=1}^{k} A_{j}) = ∑_{j=1}^{k} Z_{μ}(A_{k}) with probability 1 if A_{1},...,A_{k} are disjoint sets.

Remark. There is the following equivalent version for the definition of random orthogonal measures: The system of random variables system of random variables Z_{μ}(A), A ∈ M, μ(A) < ∞, is a Gaussian random orthogonal measure corresponding to the measure μ if

(i′) Z_{μ}(A_{1}),...,Z_{μ}(A_{k}) are (jointly) Gaussian random variables for all sets A_{j} ∈

M, μ(A_{j}) < ∞, j = 1,...,k. (ii′) EZ_{μ}(A) = 0, and EZ_{μ}(A)Z_{μ}(B) = μ(A∩B) if A, B ∈ M, μ(A) < ∞, μ(B) <

∞.

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 [Z_{μ} ( ⋃_{j=1}^{k} A_{j}) − ∑_{j=1}^{k} Z_{μ}(A_{k})]^{2} = 0

if A_{1},...,A_{k} are disjoint sets.

The second characterization of random orthogonal measures may help to show that for any measure space (M,M ,μ) with a σ-finite measure μ there exists a Gaussian random orthogonal measure corresponding to the measure μ. The main point in checking this statement is the proof that for any sets A_{1},...,A_{k} ∈ M, μ(A_{j}) < ∞, 1 ≤ j ≤ k, there exists a Gaussian random vector (Z_{μ}(A_{1}),...,Z_{μ}(A_{k})), EZ_{μ}(A_{j}) = 0, with correlation EZ_{μ}(A_{i})Z_{μ}(A_{j}) = μ(A_{i}∩ A_{j}) for all 1 ≤ i,j ≤ k. To prove this we have to show that the corresponding covariance matrix is really positive definite, i.e. ∑_{i,j} c_{i}c̄_{j}μ(A_{i}∩A_{j}) ≥ 0 for an arbitrary vector (c_{1},...,c_{k}). But this fol-

lows from the observation ∑_{i,j} c_{i}c̄_{j}χ_{A_{i}∩A_{j}}(x) = ∑_{i,j} c_{i}c̄_{j}χ_{A_{i}}(x)\overline{χ_{A_{j}}(x)}= |∑_{i} c_{i}χ_{A_{i}}(x)|^{2} ≥ 0

for all x ∈ M, if we integrate this inequality with respect to the measure μ in the space M.

We define the real Hilbert spaces K̄_{μ}^{n}, n = 1,2,.... The space K̄_{μ}^{n} consists of the real-valued measurable functions over (\underbrace{M × ··· × M}_{n times}, \underbrace{M × ··· × M}_{n times}) such that

∥f∥^{2} = ∫ |f(x_{1},...,x_{n})|^{2}μ(dx_{1})...μ(dx_{n}) < ∞,

and the last formula defines the norm in K̄_{μ}^{n}. Let K_{μ}^{n} denote the subspace of K̄_{μ}^{n} consisting of the functions f ∈ K̄_{μ}^{n} such that

f(x_{1},...,x_{n}) = f(x_{π(1)},...,x_{π(n)}) for all π ∈ Π_{n}.

Let the spaces K̄_{μ}^{0} and K_{μ}^{0} consist of the real constants with the norm ∥c∥ = |c|. Finally we define the Fock space ExpK_{μ} which consists of the sequences f = (f_{0},f_{1},...), f_{n} ∈ K_{μ}^{n}, n = 0,1,2,..., such that

∥f∥^{2} = ∑_{n=0}^{∞} \frac{1}{n!}∥f_{n}∥^{2} < ∞.

Given a random orthogonal measure Z_{μ} corresponding to μ, let us introduce the σ-algebra F = σ(Z_{μ}(A): A ∈ M , μ(A) < ∞). Let K denote the real Hilbert space of square integrable random variables measurable with respect to the σ-algebra F. Let K_{≤n} denote the subspace that is the closure of the linear space containing the polynomials of the random variables Z_{μ}(A) of order less than or equal to n. Let K_{n} be the orthogonal completion of K_{≤n−1} to K_{≤n}. (The norm is defined as ∥ξ∥^{2} = Eξ^{2} in these Hilbert spaces.)

The multiple Wiener–Itô integrals with respect to the random orthogonal measure Z_{μ}, to be defined below, give a unitary transformation from ExpK_{μ} to K . We shall denote these integrals by ∫′to distinguish them from the Wiener–Itô integrals defined in Chapter 4.

First we define the class of simple functions ˆ ¯K_{μ}^{n}⊂ K̄_{μ}^{n}. A function f ∈ K̄_{μ}^{n} is in

ˆ¯K_{μ}^{n} if there exists a finite system of disjoint sets Δ_{1},...,Δ_{N}, with Δ_{j} ∈ M, μ(Δ_{j}) < ∞, j = 1,...,N, such that f(x_{1},...,x_{n}) is constant on the sets Δ_{j_{1}} × ··· × Δ_{j_{n}} if the indices j_{1},...,j_{n} are disjoint, and f(x_{1},...,x_{n}) equals zero outside these sets. We define

∫ ′f(x_{1},...,x_{n})Z_{μ}(dx_{1})...Z_{μ}(dx_{n}) = ∑ f(x_{j_{1}},...,x_{j_{n}})Z_{μ}(Δ_{j_{1}})··· Z_{μ}(Δ_{j_{n}})

for f ∈ ˆ ¯K_{μ}^{n}, where x_{k} ∈ Δ_{k}, k = 1,...,N.

Let K̂_{μ}^{n} = ˆ ¯K_{μ}^{n}∩ K_{μ}^{n}. The random variables

I′_{μ}(f) = \frac{1}{n!} ∫ ′f(x_{1},...,x_{n})Z_{μ}(dx_{1})...Z_{μ}(dx_{n}), f ∈ ˆ ¯K_{μ}^{n},

have zero expectation, integrals of different order are orthogonal,

I′_{μ}(f) = I′_{μ}(Symf), and Symf ∈ K̂_{μ}^{n} if f ∈ ˆ ¯K_{μ}^{n},
EI′_{μ}(f)^{2} ≤ \frac{1}{n!}∥f∥^{2} if f ∈ ˆ ¯K_{μ}^{n},  (7.1)

and (7.1) holds with equality if f ∈ K̂_{μ}^{n}.

It can be seen that ˆ ¯K_{μ}^{n} is dense in K̄_{μ}^{n} in the L_{2}(μ^{n}) norm. (This is a statement analogous to Lemma 4.1, but its proof is simpler. Hence relation (7.1) enables us to extend the definition of the n-fold Wiener–Itô integrals over K̄_{μ}^{n}. All the above

mentioned relations remain valid if f ∈ ˆ ¯K_{μ}^{n} is substituted by f ∈ K̄_{μ}^{n}, and f ∈ K̂_{μ}^{n} is substituted by f ∈ K_{μ}^{n}. We formulate Itô’s formula for these integrals. It can be proved similarly to Theorem 4.3 with the help of the diagram formula.

Theorem 7.1. (Itô’s formula.) Let φ_{1},...,φ_{m}, φ_{j} ∈ K_{μ}^{1} for all 1 ≤ j ≤ m, be an orthonormal system in L_{μ}^{2}. Let some positive integers j_{1},...,j_{m} be given, put j_{1} + ··· + j_{m}= N, and define for all i = 1,...,N

g_{i} = φ_{1} for 1 ≤ i ≤ j_{1}, and g_{i} = φ_{s} for j_{1}+ ··· + j_{s−1} < i ≤ j_{1}+ ··· + j_{s}.

Then

H_{j_{1}} ( ∫ ′φ_{1}(x)Z_{μ}(dx)) ··· H_{j_{m}} ( ∫ ′φ_{m}(x)Z_{μ}(dx))
= ∫ ′g_{1}(x_{1})··· g_{N}(x_{N})Z_{μ}(dx_{1})...Z_{μ}(dx_{N})
= ∫ ′Sym[g_{1}(x_{1})··· g_{N}(x_{N})]Z_{μ}(dx_{1})...Z_{μ}(dx_{N}).

(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 μ(dx_{j}), N − 2|γ| + 1 ≤ j ≤ N − |γ|, in the definition of h_{γ} in formula (5.1).)

It can be seen with the help of Theorem 7.1 that the transformation

I′_{μ}: ExpK_{μ} → K ,

where I′_{μ}(f) = ∑_{n=0}^{∞} I′_{μ}(f_{n}), f = (f_{0},f_{1},...) ∈ ExpK_{μ} is a unitary transformation,

and so are the transformations (n!)^{1/2}I′_{μ} from K_{μ}^{n} to K_{n}.

Let us consider the special case (M,M ,μ) = (R^{ν},B^{ν},λ), where λ denotes the Lebesgue measure in R^{ν}. A random orthogonal measure corresponding to λ is called the white noise. A random spectral measure corresponding to λ, 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–Itô integrals with respect to white noise.

Proposition 7.2. Let f = (f_{0},f_{1},...,) ∈ ExpK_{λ} be an element of the Fock space corresponding to the Lebesgue measure in the Euclidean space (R^{ν},B^{ν}). Then f′ = (f′_{0},f′_{1},...,) ∈ ExpH_{λ} with the functions f′_{0} = f_{0} and f′_{n} = (2π)^{−nν/2} f̃_{n}, n = 1,2,..., (where f̃_{n}(u_{1},...,u_{n}) = ∫_{R^{nν}}e^{i(x,u)}f_{n}(x_{1},...,x_{n})dx_{1}... dx_{n} with x = (x_{1},...,x_{n}) and u = (u_{1},...,u_{n})), and

∑_{n=0}^{∞} \frac{1}{n!} ∫ ′f_{n}(x_{1},...,x_{n})Z_{λ}(dx_{1})...Z_{λ}(dx_{n})

=^{Δ} ∑_{n=0}^{∞} \frac{1}{n!} ∫ f′_{n}(u_{1},...,u_{n})Z_{λ}(du_{1})...Z_{λ}(du_{n}),

where Z_{λ}(dx) is a white noise as a random orthogonal measure, and Z_{λ}(du) is a white noise as a random spectral measure.

Proof of Proposition 7.2. We have

(2π)^{−nν/2} ∥f̃_{n}∥_{L_{λ}^{2}} = ∥f_{n}∥_{L_{λ}^{2}},

hence f′∈ ExpH_{λ}.

Let φ_{1},φ_{2},... be a complete orthonormal system in L_{λ}^{2}. Then φ′_{1},φ′_{2},... is also a complete orthonormal system in L_{λ}^{2}, and if

f_{n}(x_{1},...,x_{n}) = ∑ c_{j_{1},...,j_{n}} φ_{j_{1}}(x_{1})··· φ_{j_{n}}(x_{n}),

then

f′_{n}(u_{1},...,u_{n}) = ∑ c_{j_{1},...,j_{n}} φ′_{j_{1}}(u_{1})··· φ′_{j_{n}}(u_{n}).

Hence an application of Itô’s formula for both types of integrals, (i.e. Theorems 4.3 and 7.1) imply Proposition 7.2. ⊓ ⊔

Finally we restrict ourselves to the case ν = 1. We formulate a result which reflects a connection between multiple Wiener–Itô integrals and classical Itô integrals. Let W (t), a ≤ t ≤ b, be a Wiener process, and let us define the random orthogonal measure Z(dx) as

Z(A) = ∫ χ_{A}(x)W (dx), A ⊂ [a,b), A ∈ B^{1}.

Then we have the following

Proposition 7.3. Let f ∈ K_{λ[a,b)}^{n}, where λ[a,b) denotes the Lebesgue measure on the interval [a,b). Then

∫ ′f(x_{1},...,x_{n})Z(dx_{1})...Z(dx_{n})  (7.2)

= n!∫_{a}^{b}( ∫_{a}^{t_{n}}( ··· ( ∫_{a}^{t_{3}}( ∫_{a}^{t_{2}}f(t_{1},...,t_{n})W (dt_{1}))W (dt_{2}))...) W (dt_{n})).

Proof of Proposition 7.3. Given a function f ∈ K̂_{λ[a,b)}^{n}, let the function f̂be defined as

f̂(x_{1},...,x_{n}) = \{ \begin{matrix}f(x_{1},...,x_{n}) if x_{1} < x_{2} < ···< x_{n} \\ 0 otherwise. \end{matrix}

It is not difficult to check Proposition 7.3 for such special functions f ∈ K̂_{λ[a,b)}^{n} for

which the function f̂is the indicator function of a rectangle of the form ∏_{j=1}^{n} [a_{j},b_{j})

with constants a ≤ a_{1} < b_{1} < a_{2} < b_{2} < ···< a_{n} < b_{n} ≤ b. Here we exploit the relation I′(f) = n!I′(f̂). Beside this, we have to calculate the value of the right-hand side of formula (7.2) for such simple functions f ∈ K̂_{λ[a,b)}^{n}. A simple inductive

argument shows that it equals ∏_{j=1}^{n} [W (b_{j})−W (a_{j})] if a ≤ a_{1} < b_{1} < a_{2} < b_{2} < ···<

a_{n} < b_{n} ≤ b, and it equals zero otherwise. Then a simple limiting procedure with the help of the approximation of general functions in K_{λ[a,b)}^{n}by the linear combinations of such functions proves Proposition 7.3 in the general case. ⊓ ⊔

As a consequence of Proposition 7.3 in the case ν = 1 multiple Wiener–Itô integrals can be substituted by Itô integrals in the investigation of most problems. In the case ν = 2 there is no simple definition of Itô integrals. On the other hand, no problem arises in generalizing the definition of multiple Wiener–Itô integrals to the case ν ≥ 2.

Non-central limit theorems

In this chapter we investigate the problem formulated in Chapter 1, and we show how the technique of Wiener–Itô integrals can be applied for the investigation of such a problem. We restrict ourselves to the case of discrete fields, although the case of generalized 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 results will be omitted. They can be found in [9]. First we recall the following

Definition 8A. (Definition of slowly varying functions.) A function L(t), t ∈ [t_{0},∞), t_{0} > 0, is said to be a slowly varying function (at infinity) if

lim_{t→∞} \frac{L(st)}{L(t)}= 1 for all s > 0.

We shall apply the following description of slowly varying functions.

Theorem 8A. (Karamata’s theorem.) If a slowly varying function L(t) is bounded on every finite interval, then it can be represented in the form

L(t) = a(t)exp\{ ∫_{t_{0}}^{t} \frac{ε(s)}{s}ds\} ,

where a(t) → a_{0} ≠ 0, and ε(t) → 0 as t → ∞, and the functions a(·) and ε(·) are bounded in every finite interval.

Let X_{n}, n ∈ ℤ_{ν}, be a stationary Gaussian field with expectation zero and a correlation function

r(n) = EX_{0}X_{n} = |n|^{−α} a( \frac{n}{|n|}) L(|n|), n ∈ ℤ_{ν},  (8.1)

where 0 < α < ν, L(t) is a slowly varying function, bounded in all finite intervals, and a(t) is a continuous function on the unit sphere S_{ν−1} , satisfying the symmetry property a(x) = a(−x) for all x ∈ S_{ν−1} . Let G denote the spectral measure of the field X_{n}, and let us define the measures G_{N}, N = 1,2,..., by the formula

G_{N}(A) = \frac{N^{α}}{L(N)}G( \frac{A}{N}) , A ∈ B^{ν}, N = 1,2,....  (8.2)

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

Definition of vague convergence of measures. Let G_{n}, n = 1,2,..., be a sequence of locally finite measures over R^{ν}, i.e. let G_{n}(A) < ∞ for all measurable bounded sets A. We say that the sequence G_{n} vaguely converges to a locally finite measure G_{0} (in notation G_{n} →^{v} G_{0}) if

lim_{n→∞} ∫ f(x)G_{n}(dx) = ∫ f(x)G_{0}(dx)

for all continuous functions f with a bounded support.

We formulate the following

Lemma 8.1. Let G be the spectral measure of a stationary random field with a correlation function r(n) of the form (8.1). Then the sequence of measures G_{N} defined in (8.2) tends vaguely to a locally finite measure G_{0}. The measure G_{0} has the homogeneity property

G_{0}(A) = t^{−α} G_{0}(tA) for all A ∈ B^{ν} and t > 0,  (8.3)

and it satisfies the identity

2^{ν} ∫ e^{i(t,x)}∏_{j=1}^{ν} \frac{1 − cosx^{(j)}}{(x^{(j)})^{2}} G_{0}(dx)  (8.4)

= ∫_{[−1,1]^{ν}} (1 − |x^{(1)}|)··· (1 − |x^{(ν)}|)\frac{a( \frac{x+t}{|x+t|})}{|x + t|^{α}} dx, for all t ∈ R^{ν}.

We postpone the proof of Lemma 8.1 for a while.

Formulae (8.3) and (8.4) imply that the function a(t) and the number α in the definition (8.1) of a correlation function r(n) uniquely determine the measure G_{0}.

Indeed, by formula (8.4) they determine the (finite) measure ∏_{j=1}^{ν} \frac{1−cosx^{(j)}}{(x^{(j)})^{2}} G_{0}(dx),

since they determine its Fourier transform. Hence they also determine the measure G_{0}. (Formula (8.3) shows that this is a locally finite measure). Let us also remark that since G_{N}(A) = G_{N}(−A) for all N = 1,2,... and A ∈ B^{ν}, the relation G_{0}(A) = G_{0}(−A), A ∈ B^{ν} 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 Theorem 8.2. Let X_{n}, n ∈ ℤ_{ν}, be a stationary Gaussian field with a correlation function r(n) satisfying relation (8.1). Let us define the stationary random field ξ_{j} = H_{k}(X_{j}), j ∈ ℤ_{ν}, with some positive integer k, where H_{k}(x) denotes the k-th Hermite polynomial with leading coefficient 1, and assume that the parameter α appearing in (8.1) satisfies the relation 0 < α < \frac{ν}{k}. If the random fields Z_{n}^{N}, N = 1,2,..., n ∈ ℤ_{ν}, are defined by formula (1.1) with A_{N} = N^{ν−kα/2} L(N)^{k/2} and the above defined ξ_{j} = H_{k}(X_{j}), then their multi-dimensional distributions tend to those of the random field Z_{n}^{∗},

Z_{n}^{∗} = ∫ χ̃_{n}(x_{1}+ ··· + x_{k})Z_{G_{0}}(dx_{1})...Z_{G_{0}}(dx_{k}), n ∈ ℤ_{ν}.

Here Z_{G_{0}} is a random spectral measure corresponding to the spectral measure G_{0} which appeared in Lemma 8.1. The function χ̃_{n}(·), n = (n^{(1)},...,n^{(ν)}), is (similarly to Chapter 6) the Fourier transform of the indicator function of the ν-dimensional

unit cube ∏_{p=1}^{ν} [n^{(p)},n^{(p)} + 1].

Remark. The condition that the correlation function r(n) of the random field X_{n}, n ∈ ℤ_{ν}, satisfies formula (8.1) can be weakened. Theorem 8.2 and Lemma 8.1 remain valid if (8.1) is replaced by the slightly weaker condition

lim_{T →∞} sup_{n: n∈ℤ_{ν},|n|≥T} \frac{r(n)}{|n|^{−α} a( \frac{n}{|n|}) L(|n|)}= 1,

where 0 < α < ν, L(t) is a slowly varying function, bounded in all finite intervals, and a(t) is a continuous function on the unit sphere S_{ν−1} , satisfying the symmetry property a(x) = a(−x) for all x ∈ S_{ν−1} .

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}(ξ)H_{k}(η) = E: ξ^{k}: : η^{k}: = k!(Eξη)^{k} for a Gaussian random vector (ξ,η) with Eξ = Eη = 0 and Eξ^{2} = Eη^{2} = 1. Hence

E(Z_{n}^{N})^{2} = \frac{k!}{A_{N}^{2}} ∑_{j,l∈B_{0}^{N}} r(j − l)^{k} ∼ \frac{k!}{A_{N}^{2}} ∑_{j,l∈B_{0}^{N}} |j − l|^{−kα} a^{k}( \frac{j − l}{|j − l|} ) L(|j − l|)^{k},

with the set B_{0}^{N} introduced after formula (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 kα < ν, 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 const.N^{2ν} such terms.

The field ξ_{n} is subordinated to the Gaussian field X_{n}. It is natural to rewrite it in canonical form, and to express Z_{n}^{N} via multiple Wiener–Itô integrals. Itô’s formula yields the relation

ξ_{n} = H_{k}( ∫ e^{i(n,x)}Z_{G}(dx)) = ∫ e^{i(n,x_{1}+··· +x_{k})} Z_{G}(dx_{1})...Z_{G}(dx_{k}),

where Z_{G} is the random spectral measure adapted to the random field X_{n}. Then

Z_{n}^{N} = \frac{1}{A_{N}} ∑_{j∈B_{n}^{N}} ∫ e^{i(j,x_{1}+··· +x_{k})} Z_{G}(dx_{1})...Z_{G}(dx_{k})

= \frac{1}{A_{N}}∫ e^{i(Nn,x_{1}+··· +x_{k})} ∏_{j=1}^{ν} \frac{e^{iN(x_{1}^{(j)}+··· +x_{k}^{(j)})} − 1}{e^{i(x_{1}^{(j)}+··· +x_{k}^{(j)})} − 1} Z_{G}(dx_{1})...Z_{G}(dx_{k}).

Let us make the substitution y_{j} = Nx_{j}, j = 1,...,k, in the last formula, and let us rewrite it in a form resembling formula (6.7). To this end, let us introduce the measures G_{N} defined in (8.2). By Lemma 4.6 we can write

Z_{n}^{N} =^{Δ} ∫ f_{N}(y_{1},...,y_{k})χ̃_{n}(y_{1}+ ··· + y_{k})Z_{G_{N}}(dy_{1})...Z_{G_{N}}(dy_{k})

with

f_{N}(y_{1},...,y_{k}) = ∏_{j=1}^{ν} \frac{i(y_{1}^{(j)} + ··· + y_{k}^{(j)})}{( exp\{i\frac{1}{N}(y_{1}^{(j)} + ··· + y_{k}^{(j)})\} − 1) N},  (8.5)

where χ̃_{n}(·) is the Fourier transform of the indicator function of the unit cube

∏_{j=1}^{ν} [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)} + ···+ y_{k}^{(j)} = 2lNπ with some integer l ≠ 0 and 1 ≤ j ≤ ν has 0 G_{N} × ···× 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 → ∞ 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–Itô integral expressing Z_{n}^{N}. We want to justify this formal limiting procedure. For this we have to show that the Wiener–Itô integral expressing Z_{n}^{N} is essentially concentrated in a large bounded region independent of N. The L_{2} isomorphism of Wiener–Itô integrals can help us in showing that. The next result formulated in Lemma 8.3 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–Itô 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 (8.1), then its spectral measure is non-atomic.

Lemma 8B. Let the correlation function of a stationary random field X_{n}, n ∈ ℤ_{ν}, satisfy the relation r(n) ≤ A|n|^{−α} with some A > 0 and α > 0 for all n ∈ ℤ_{ν}, n ≠ 0.

Then its spectral measure G is non-atomic. Moreover, all hyperplanes ∑_{j=1}^{ν} c_{j}x^{(j)} = d

defined with some constants c_{j} and d have zero G measure.

Proof of Lemma 8B. Lemma 8B clearly holds if α > ν, because in this case the spectral measure G has even a density function g(x) = ∑_{n∈ℤ_{ν}} 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^{ν}/2πℤ_{ν}) has Fourier transform, r(n)^{p}, n ∈ Z^{ν} hence in the case p > \frac{ν}{α} this function is non-atomic. Hence it is enough to show that if the convolution G ∗ G is a non-atomic measure, then so is the measure G. But this is obvious, because if there were a point x ∈ R^{ν}/2πℤ_{ν} such that G({x}) > 0, then G ∗ G({x + x}) > 0 would hold, and this is a contradiction. (Here addition is taken on the torus.) It can be proved similarly that all hyperplanes have zero G measure. ⊓ ⊔

Now we formulate the following result.

Lemma 8.3. Let G_{N}, N = 1,2,..., be a sequence of non-atomic spectral measures on R^{ν} tending vaguely to a non-atomic spectral measure G_{0}. Let a sequence of measurable functions K_{N} = K_{N}(x_{1},...,x_{k}), N = 0,1,2,..., be given such that K_{N} ∈ H̄_{G_{N}}^{k} for N = 1,2,.... Assume further that the following properties hold: For all ε > 0 there exist some constants A = A(ε) > 0 and N_{0} = N_{0}(ε) > 0 and finitely many rectangles P_{1},...,P_{M} with some cardinality M = M(ε) on R^{kν} which satisfy the following conditions (a) and (b) formulated below. (We call a set P ∈ B^{kν} a rectangle if it can be written in the form P = L_{1}× ··· × L_{k} with some bounded open sets L_{s} ∈ B^{ν}, 1 ≤ s ≤ k, with boundaries ∂L_{s} of zero G_{0} measure, i.e. G_{0}(∂L_{s}) = 0 for all 1 ≤ s ≤ k.)

(a) The function K_{0} is continuous on the set B = [−A,A]^{kν} \ ⋃_{j=1}^{M} P_{j}, and K_{N} → K_{0}

uniformly on the set B as N → ∞. Besides, the hyperplanes x_{p} = ±A have zero G_{0} measure for all 1 ≤ p ≤ ν.

(b) ∫_{R^{kν}\B}|K_{N}(x_{1},...,x_{k})|^{2}G_{N}(dx_{1})...G_{N}(dx_{k}) < \frac{ε^{3}}{k!} if N = 0 or N ≥ N_{0}, and K_{0}(−x_{1},...,−x_{k}) = \overline{K_{0}(x_{1},...,x_{k})}for all (x_{1},...,x_{k}) ∈ R^{kν}.

Then K_{0} ∈ H̄_{G_{0}}^{k} , and

∫ K_{N}(x_{1},...,x_{k})Z_{G_{N}}(dx_{1})...Z_{G_{N}}(dx_{k}) →^{D} ∫ K_{0}(x_{1},...,x_{k})Z_{G_{0}}(dx_{1})...Z_{G_{0}}(dx_{k})

as N → ∞, where →^{D} denotes convergence in distribution.

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–Itô 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 seemed to be useful to introduce some metric in the space of probability measures which metricizes weak convergence. Although it may look a bit too technical, it made possible to carry out some arguments in a natural way.

Proof of Lemma 8.3. Conditions (a) and (b) obviously imply that

∫ |K_{0}(x_{1},...,x_{k})|^{2}G_{0}(dx_{1})...G_{0}(dx_{k}) < ∞,

hence K_{0} ∈ H̄_{G_{0}}^{k} . Let us fix an ε > 0, and let us choose some A > 0, N_{0} > 0 and rectangles P_{1},...,P_{M} which satisfy conditions (a) and (b) with this ε. Then

E [∫ [1 − χ_{B}(x_{1},...,x_{k})]K_{N}(x_{1},...,x_{k})Z_{G_{N}}(dx_{1})...Z_{G_{N}}(dx_{k})]^{2}
≤ k!∫_{R^{kν}\B}|K_{N}(x_{1},...,x_{k})|^{2}G_{N}(dx_{1})...G_{N}(dx_{k}) < ε^{3}  (8.6)

for N = 0 or N > N_{0}, where χ_{B}denotes the indicator function of the set B introduced in the formulation of condition (a).

Since B ⊂ [−A,A]^{kν}, and G_{N} →^{v} G_{0}, hence G_{N} × ···× G_{N}(B) < C(A) with an

appropriate constant C(A) < ∞ for all N = 0,1,.... Because of this estimate and the uniform convergence K_{N} → K_{0} on the set B we have

E [∫ (K_{N}(x_{1},...,x_{k}) − K_{0}(x_{1},...,x_{k}))χ_{B}(x_{1},...,x_{k})Z_{G_{N}}(dx_{1})...Z_{G_{N}}(dx_{k})]^{2}

≤ k!∫_{B}|K_{N}(x_{1},...,x_{k}) − K_{0}(x_{1},...,x_{k})|^{2}G_{N}(dx_{1})...G_{N}(dx_{k}) < ε^{3} (8.7)

for N > N_{1} with some N_{1} = N_{1}(A,ε).

First we shall reduce the proof of Lemma 8.3 to the proof of the relation

∫ K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k})Z_{G_{N}}(dx_{1})...Z_{G_{N}}(dx_{k})
→^{D} ∫ K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k})Z_{G_{0}}(dx_{1})...Z_{G_{0}}(dx_{k}). (8.8)

with the help of formulas (8.6) and (8.7), and then we shall prove (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.

Definition of Prokhorov metric, and its properties. Given a separable metric space (X,ρ) with some metric ρ let S denote the space of probability measures on it. The Prokhorov metric ρ_{P} is the metric in the space S defined by the formula ρ_{P}(μ,ν) = inf{ε: μ(A) ≤ ν(A^{ε}) + ε for all A ∈ A } for two probability measures μ,ν ∈ S, where A^{ε} = {x: ρ(x,A) < ε}. The above defined ρ_{P} is really a metric on S (in particular, ρ_{P}(μ,ν) = ρ_{P}(ν,μ)) which metricizes the weak convergence of probability measures in the metric space (X,ρ), i.e. μ_{N} →^{w} μ_{0} for a sequence of probability measures N = 0,1,2,... if and only if lim_{N→∞} ρ_{P}(μ_{N},μ_{0}) = 0.

The results formulated in this definition can be found e.g. in 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.

Definition of weak convergence of probability measures on a metric space. A sequence of probability measures μ_{n}, n = 1,2,..., on a metric space (X,ρ) converges weakly to a probability measure μ on this space, (in notation μ_{n} →^{w} μ) if lim_{n→∞} ∫ f(x)μ_{n}(dx) → ∫ f(x)μ(dx) for all continuous and bounded functions on the space (X,ρ).

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 ξ let μ(ξ) denote its distribution. Let us remark that the convergence ξ_{N} →^{D} ξ_{0} as N → ∞ of a sequence of random variables, ξ_{0},ξ_{1},ξ_{2},... is equivalent to the statement μ(ξ_{N}) →^{w} μ(ξ_{0}) or ρ_{P}(μ(ξ_{N}),μ(ξ_{0})) → 0 as N → ∞. Hence by putting ξ_{N} = k!I_{G_{N}}(K_{N}(x_{1},...,x_{k})), N = 0,1,2,... we can reformulate the statement of Lemma 8.3 in the following way. For all ε > 0 there exists some index N′_{0}= N′_{0}(ε) such that ρ_{P}(μ(ξ_{N}),μ(ξ_{0})) ≤ 4ε for all N ≥ N′_{0}.

To prove the reduction of Lemma 8.3 to formula (8.8) let us first show that for three random variables ξ, ξ̄ and η such that P(|η| ≥ ε) ≤ ε the inequality

ρ_{P}(μ(ξ + η),μ(ξ̄)) ≤ ρ_{P}(μ(ξ),μ(ξ̄)) + ε  (8.9)

holds.

Indeed, since {ω: ξ(ω) + η(ω) ∈ A} ⊂ {ω: ξ(ω) ∈ A^{ε}} ∪ {ω: |η(ω)| ≥ ε}, we have P(ξ + η ∈ A) ≤ P(ξ ∈ A^{ε}) + ε for any set A ∈ B_{1} if P(|η| ≥ ε) ≤ ε. Besides, P(ξ ∈ A^{ε}) ≤ P(ξ̄ ∈ A^{ε+δ} ) + δ for all δ > ρ_{P}(μ(ξ),μ(ξ̄)). Hence P(ξ + η ∈ A) ≤ P(ξ̄ ∈ A^{ε+δ} ) + ε + δ for all A ∈ B_{1} and δ > ρ_{P}(μ(ξ),μ(ξ̄)), i.e. ρ_{P}(μ(ξ +η),μ(ξ̄)) ≤ ε +δ, and this implies the inequality ρ_{P}(μ(ξ +η),μ(ξ̄)) ≤ ρ_{P}(μ(ξ),μ(ξ̄)) + ε.

Put

ξ_{N}^{(1)} = k!I_{G_{N}}(K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k})),
ξ_{N}^{(2)} = k!I_{G_{N}}(K_{N}(x_{1},...,x_{k}) − K_{0}(x_{1},...,x_{k}))χ_{B}(x_{1},...,x_{k})),
ξ_{N}^{(3)} = k!I_{G_{N}}(1 − χ_{B}(x_{1},...,x_{k}))K_{N}(x_{1},...,x_{k}))

for all N = 0,1,2,.... With this notation it follows from relation (8.8) and the fact that the Prokhorov metric metricizes the weak convergence that

ρ_{P}(μ(ξ_{N}^{(1)}),μ(ξ_{0}^{(1)})) ≤ ε if N ≥ N′_{1}(ε)

with some threshold index N′_{1}(ε). Formulas (8.6) and (8.7) together with the Chebishev inequality imply that P(|ξ_{N}^{(2)}| ≥ ε) ≤ ε and P(|ξ_{N}^{(3)}| ≥ ε) ≤ ε if N ≥ N′_{2}(ε) or N = 0 with some threshold index N′_{2}(ε). Besides, we have ξ_{0} = ξ_{0}^{(1)} + ξ_{0}^{(3)} and ξ_{N} = ξ_{N}^{(1)} + ξ_{N}^{(2)} + ξ_{N}^{(3)} for N = 1,2,.... The above mentioned properties of the random variables we considered together with relation (8.9) imply that

ρ_{P}(μ(ξ_{N}),μ(ξ_{0})) = ρ_{P}(μ(ξ_{N}^{(1)}+ ξ_{N}^{(2)}+ ξ_{N}^{(3)}),μ(ξ_{0}^{(1)}+ ξ_{0}^{(3)}))
≤ ρ_{P}(μ(ξ_{N}^{(1)}+ ξ_{N}^{(2)}+ ξ_{N}^{(3)}),μ(ξ_{0}^{(1)})) + ε
≤ ρ_{P}(μ(ξ_{N}^{(1)}+ ξ_{N}^{(2)}),μ(ξ_{0}^{(1)})) + 2ε
≤ ρ_{P}(μ(ξ_{N}^{(1)}),μ(ξ_{0}^{(1)})) + 3ε ≤ 4ε

if N ≥ N′_{0}(ε) = max(N′_{1}(ε),N′_{2}(ε)). Hence Lemma 8.3 follows from (8.8).

To prove (8.8) we will show that K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k}) can be well approximated by simple functions from ˆ ¯H_{G_{0}}^{k} in the following sense. For all ε′> 0 there

exists a simple function f_{ε′}∈ ˆ ¯H_{G_{0}}^{k} such that

E ∫ (K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k}) − f_{ε′}(x_{1},...,x_{k}))^{2}G_{0}(dx_{1})...G_{0}(dx_{k}) ≤ \frac{ε′^{3}}{k!}

(8.10)

and also

E ∫ (K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k}) − f_{ε′}(x_{1},...,x_{k}))^{2}G_{N}(dx_{1})...G_{N}(dx_{k}) ≤ \frac{ε′^{3}}{k!}

(8.11)

if N ≥ N_{0} with some threshold index N_{0} = N_{0}(ε′,K_{0}(·)χ_{B}(·)). Moreover, this simple function f_{ε′}is adapted to such a regular system D = {Δ_{j}, j = ±1,...,±M} whose elements have boundaries with zero G_{0} measure, i.e. G_{0}(∂Δ_{j}) = 0 for all 1 ≤ |j| ≤ M.

To prove (8.8) with the help of these estimates first I show that this function f_{ε′}∈ ˆ ¯H_{G_{0}}^{k} satisfies the relation

∫ f_{ε′}(x_{1},...,x_{k})Z_{G_{N}}(dx_{1})...Z_{G_{N}}(dx_{k}) →^{D} ∫ f_{ε′}(x_{1},...,x_{k})Z_{G_{0}}(dx_{1})...Z_{G_{0}}(dx_{k})

(8.12)

as N → ∞. To prove (8.12) observe that for the regular system D = {Δ_{j}, j = ±1,...,±M} to which the function f_{ε′}∈ ˆ ¯H_{G_{0}}^{k} is adapted has the property G_{0}(∂Δ_{j}) = 0 for all j = ±1,...,±M. Besides, the spectral measures G_{N} are such that G_{N} →^{v} G_{0}.

Hence the (Gaussian) random vectors (Z_{G_{N}}(Δ_{j}), j = ±1,...,±M) converge in distribution to the (Gaussian) random vector (Z_{G_{0}}(Δ_{j}), j = ±1,...,±M) as N → ∞. The same can be told about the random variables we get by putting the arguments of these random vectors to a continuous function (of M variables). Since the integrals in (8.12) are polynomials of these random vectors, we can apply these results for them, and they imply relation (8.12).

Put

K_{0}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k}) − f_{ε′}(x_{1},...,x_{k}) = h_{0}(x_{1},...,x_{k}).  (8.13)

By relations (8.10), (8.11) and the Chebishev inequality P(|k!I_{G_{0}}(h_{0})| ≥ ε′) ≤ ε′ and P(|k!I_{G_{N}}(h_{0}) ≥ ε′) ≤ ε′ if N ≥ N_{0}. Since I_{G_{N}}(K_{N}(x_{1},...,x_{k})χ_{B}(x_{1},...,x_{k})) = I_{G_{N}}(f_{ε′}(x_{1},...,x_{k})) + h_{N}(x_{1},...,x_{k}), N = 0,1,2,..., the above relations together with formulas (8.12) and (8.9) (with the number ε′instead of ε) imply that

lim_{N→∞} ρ_{P}(μ(k!I_{G_{N}}(K_{0}(·)χ_{B}(·))),μ(k!I_{G_{0}}(K_{0}(·)χ_{B}(·))))

= lim_{N→∞} ρ_{P}(μ(k!I_{G_{N}}(f_{ε′}(·) + h_{0}(·))),μ(k!I_{G_{o}}(f_{ε′}(·) + h_{0}(·))))

≤ lim_{N→∞} ρ_{P}(μ(k!I_{G_{N}}(h_{0}(·))),μ(k!I_{G_{0}}(h_{0}(·)))) + 2ε′= 2ε′.

Since this inequality holds for all ε′> 0 this implies relation (8.8). To complete the proof of Lemma 8.3 we have to justify relations (8.10) and (8.11).

Relation (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 ε′the function K_{0}(·)χ_{B}(·) can be approximated with a simple function f_{ε′}(x_{1},...,x_{k}) which satisfies (8.10) agrees with Lemma 4.1. But now we want to find a function f_{ε′}which is adapted to such a regular system D = {Δ_{j}, j = ±1,...,±M} whose elements have the additional property G_{0}(∂Δ_{j}) = 0 for all indices j. A function f_{ε′}with these properties can be constructed by means of a slight modification of the proof of Lemma 4.1. But in the present case we exploit that the function K_{0}(·)χ_{B}(·) is almost everywhere continuous with respect to the product measure G_{0}^{k} = \underbrace{G_{0}× ··· × G_{0}}_{k 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 Statement A. In this modified version we claim the good approximability of the indicator function of such sets A which satisfy not only the properties demanded in Statement A, but also the relations G_{0}(∂A) = 0 and G_{0}(∂A_{1}) = 0 hold. On the other hand, we demand the same property G_{0}(∂B) = 0 about the set B whose indicator function is the approximating function in Statement A. To carry out the reduction, needed in this case we approximate the function K_{0}(·)χ_{B}(·) with such an elementary function (a function taking finitely many values) whose level sets have boundaries with zero G_{0}^{k} = G_{0} × ··· × G_{0} measure. This is possible, since the boundaries of these level sets consist of such points where either the function K_{0}(·)χ_{B}(·) takes the value from an appropriately chosen finite set, or it is discontinuous.

To complete the reduction of our result to the new version of Statement A we still have to show that if the set A can be written in the form A = A_{1} ∪ (−A_{1}) such that A_{1}∩(−A_{1}) = ∅, and G_{0}^{k}(∂A_{1}) = 0, then for all η > 0 there is some Ā_{1}= Ā_{1}(η) ⊂ A_{1} such that G_{0}^{k}(A \ (Ā_{1}∪ (−Ā_{1})) ≤ η, ρ(Ā_{1},−Ā_{1}) > 0, and G_{0}^{k}(∂Ā_{1}) = 0. Indeed, there is a compact set K ⊂ A_{1} such that G^{k}(A_{1} ⊂ K) ≤ \frac{η}{2}. Then also the relation ρ(K,−K) = δ > 0 holds. By the Heine–Borel theorem we can find an open set G such that K ⊂ G ⊂ K^{δ/3} with K^{δ/3} = \{x: ρ(x,K) < \frac{δ}{3}\}, and G_{0}^{k}(∂G) = 0. Then the set Ā_{1}= A_{1}∩ G satisfies the desired properties.

After making the reduction of the result we want to prove to this modified version of Statement A we can follow the construction of Lemma 4.1, but we choose in each step sets with zero G_{0}× ··· × G_{0} boundary.

A more careful analysis shows that the function constructed in such a way satisfies also (8.11) for N ≥ N_{0} with a sufficiently large threshold index N_{0}. Here we exploit that G_{N} →^{v} G_{0}. This may enable 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 (8.11) by exploiting that the function h_{0}(x_{1},...,x_{k}) defined in (8.13) is almost everywhere continuous with respect to the measure G_{0} × ···× G_{0}. It can be shown that the vague convergence has similar properties as the weak convergence, hence the above mentioned almost everywhere continuity implies that

lim_{N→∞} ∫ h_{0}(x_{1},...,x_{k})G_{N}(dx_{1})...G_{N}(dx_{k}) = ∫ h_{0}(x_{1},...,x_{k})G_{0}(dx_{1})...G_{0}(dx_{k}).

⊓ ⊔

Remark. We have formulated this statement in the case when G_{N} is a spectral measure on R^{ν}. But it remains valid if G_{N} is a spectral measure on the torus of size 2C_{N}π with C_{N} → ∞ if N → ∞ if we identify this torus with the set [−C_{N}π,C_{N}π)^{ν} ⊂ R^{ν} in a natural way.

Now we turn to the proof of Theorem 8.2.

Proof of Theorem 8.2. We want to prove that for all positive integers p, real numbers c_{1},...,c_{p} and n_{l} ∈ ℤ_{ν}, l = 1,...,p,

∑_{l=1}^{p} c_{l}Z_{n_{l}}^{N} →^{D} ∑_{l=1}^{p} c_{l}Z_{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

∑_{l=1}^{p} c_{l}Z_{n_{l}}^{N} = \frac{1}{A_{N}} ∑_{l=1}^{p} c_{l}∫ ∑_{j∈B_{n_{l}}^{N}} e^{i(j,x_{1}+··· +x_{k})} Z_{G}(dx_{1})...Z_{G}(dx_{k}),

and

∑_{l=1}^{p} c_{l}Z_{n_{l}}^{N} =^{Δ} ∫ K_{N}(x_{1},...,x_{k})Z_{G_{N}}(dx_{1})...Z_{G_{N}}(dx_{k})

with

K_{N}(x_{1},...,x_{k})= \frac{1}{N^{ν}} ∑_{l=1}^{p} c_{l} ∑_{j∈B_{n_{l}}^{N}} exp\{ i( \frac{j}{N},x_{1}+ ··· + x_{k}) \}

= f_{N}(x_{1},...,x_{k})∑_{l=1}^{p} c_{l}χ̃_{n_{l}}(x_{1}+ ··· + x_{k}).  (8.14)

with the function f_{N} defined in (8.5) and the measure G_{N} defined in (8.2), where χ̃_{n}(·) denotes the Fourier transform of the indicator function of the unit cube

∏_{j=1}^{ν} [n^{(j)},n^{(j)} + 1), n = (n^{(1)},...n^{(ν)}).

Let us define the function

K_{0}(x_{1},...,x_{k}) = ∑_{l=1}^{p} c_{l}χ̃_{n_{l}}(x_{1}+ ··· + x_{k})

and the measures μ_{N} on R^{kν} by the formula

μ_{N}(A) = ∫_{A}|K_{N}(x_{1},...,x_{k})|^{2}G_{N}(dx_{1})...G_{N}(dx_{k}),
A ∈ B^{kν} and N = 0,1,...,  (8.15)

where G_{0} is the vague limit of the measures G_{N}.

To prove Theorem 8.2 it is enough to show 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} →^{v} G_{0}, and K_{N} → K_{0} uniformly in all bounded regions in R^{kν}, it is enough to show, beside the proof of Lemma 8.1, that the measures μ_{N}, N = 1,2,..., tend weakly to the (necessary finite) measure μ_{0} which is also defined in (8.15), (in notation μ_{N} →^{w} μ_{0}), i.e. ∫ f(x)μ_{N}(dx) → ∫ f(x)μ_{0}(dx) for all continuous and bounded functions f on R^{kν}. 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 μ̄_{0} such that μ_{N} →^{w} μ̄_{0}, since then μ̄_{0} must coincide with μ_{0} because of the relations G_{N} →^{v} G_{0} and K_{N} → K_{0} uniformly in all bounded regions of R^{kν}, and K_{0} is a continuous function.

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 μ_{N} →^{w} μ̄_{0}. On the other hand, we have the additional information that the measures μ_{N}, N = 1,2,..., are concentrated in the cubes [−Nπ,Nπ)^{kν}, since the spectral measure G is concentrated in [−π,π)^{ν}. It is more fruitful to apply a version of the above mentioned theorem, where we can exploit our additional information. We formulate the following

Lemma 8.4. Let μ_{1},μ_{2},... be a sequence of finite measures on R^{l} such that μ_{N}(R^{l} \ [−C_{N}π,C_{N}π)^{l}) = 0 for all N = 1,2,..., with some sequence C_{N} → ∞ as N → ∞. Define the modified Fourier transform

φ_{N}(t) = ∫_{R^{l}}exp\{ i( \frac{[tC_{N}]}{C_{N}},x) \} μ_{N}(dx), t ∈ R^{l},

where [tC_{N}] is the integer part of the vector tC_{N} ∈ R^{l}. (For an x ∈ R^{l} its integer part [x] is the vector n ∈ ℤ_{l} for which x^{(p)}−1 < n^{(p)} ≤ x^{(p)} if x^{(p)} ≥ 0, and x^{(p)} ≤ n^{(p)} < x^{(p)} + 1 if x^{(p)} < 0 for all p = 1,2,...,l.) If for all t ∈ R^{l} the sequence φ_{N}(t) tends to a function φ(t) continuous at the origin, then the measures μ_{N} weakly tend to a finite measure μ_{0}, and φ(t) is the Fourier transform of μ_{0}.

I make some comments on the conditions of Lemma 8.4. Let us observe that if the measures μ_{N} or a part of them are shifted with a vector 2πC_{N}u with some u ∈ ℤ_{l}, then their modified Fourier transforms φ_{N}(t) do not change because of the periodicity of the trigonometrical functions e^{i(j/C_{N},x)}, j ∈ ℤ_{l}. On the other hand, these new measures which are not concentrated in [−C_{N}π,C_{N}π)^{l}, have no limit. Lemma 8.4 states that if the measures μ_{N} are concentrated in the cubes [−C_{N}π,C_{N}π)^{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.

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 ε > 0 there exits some K = K(ε) such that

μ_{N}(x: x ∈ R^{l}, |x^{(1)}| > K) < ε for all N ≥ 1.  (8.16)

As φ(t) is continuous at the origin there is some δ > 0 such that

|φ(0,...,0) − φ(t,0,...,0)| < \frac{ε}{2} if |t| < δ.  (8.17)

We have

0 ≤ Re[φ_{N}(0,...,0) − φ_{N}(t,0,...,0)] ≤ 2φ_{N}(0,...,0)  (8.18)

for all N = 1,2,.... The sequence in the middle term of (8.18) tends to

Re[φ(0,...,0) − φ(t,0,...,0)]

as N → ∞. The right-hand side of (8.18) is a bounded function in the variable N, since it is convergent. Hence the dominated convergence theorem can be applied. We get because of the condition C_{N} → ∞ and relation (8.17) that

lim_{N→∞} ∫_{0}^{[δC_{N}]/C_{N}}\frac{1}{δ} Re[φ_{N}(0,...,0) − φ_{N}(t,0,...,0)]dt

= ∫_{0}^{δ} \frac{1}{δ} Re[φ(0,...,0) − φ(t,0,...,0)]dt < \frac{ε}{2}

with this δ > 0. Hence

\frac{ε}{2} > lim_{N→∞} ∫_{0}^{[δC_{N}]/C_{N}}\frac{1}{δ} Re[φ_{N}(0,...,0) − φ_{N}(t,0,...,0)]dt
= lim_{N→∞} ∫ ( \frac{1}{δ} ∫_{0}^{[δC_{N}]/C_{N}}Re[1 − e^{i[tC_{N}]x^{(1)}/C_{N}}]dt) μ_{N}(dx)

= lim_{N→∞} ∫ \frac{1}{δC_{N}} ∑_{j=0}^{[δC_{N}]−1}Re [1 − e^{ijx^{(1)}/C_{N}}] μ_{N}(dx)

≥ limsup_{N→∞}∫_{\{|x^{(1)}|>K\}} \frac{1}{δC_{N}} ∑_{j=0}^{[δC_{N}]−1}Re,[1 − e^{ijx^{(1)}/C_{N}}] μ_{N}(dx)

= limsup_{N→∞}∫_{\{|x^{(1)}|>K\}} ( 1 − \frac{1}{δC_{N}}Re \frac{1 − e^{i[δC_{N}]x^{(1)}/C_{N}}}{1 − e^{ix^{(1)}/C_{N}}}) μ_{N}(dx)

with an arbitrary K > 0. (In the last but one step of this calculation we have exploited

that \frac{1}{δC_{N}} ∑_{j=0}^{[δC_{N}]−1}Re[1 − e^{ijx^{(1)}/C_{N}}] ≥ 0 for all x^{(1)} ∈ R^{1}.)

Since the measure μ_{N} is concentrated in {x: x ∈ R^{l}, |x^{(1)}| ≤ C_{N}π}, and

Re \frac{1 − e^{i[δC_{N}]x^{(1)}/C_{N}}}{1 − e^{ix^{(1)}/C_{N}}} = \frac{Re ( ie^{−ix^{(1)}/2C_{N}} ( 1 − e^{i[δC_{N}]x^{(1)}/C_{N}}) )}{i(e^{−ix^{(1)}/2CN} − e^{ix^{(1)}/2C_{N}})}
≤ \frac{1}{|sin( \frac{x^{(1)}}{2C_{N}}) |} ≤ \frac{C_{N}π}{|x^{(1)}|}

if |x^{(1)}| ≤ C_{N}π, (here we exploit that |sinu| ≥ \frac{2}{π}|u| if |u| ≤ \frac{π}{2}), hence we have with the choice K = \frac{2π}{δ}

\frac{ε}{2} > limsup_{N→∞}∫_{\{|x^{(1)}|>K\}} ( 1 − | \frac{π}{δx^{(1)}}|) μ_{N}(dx) ≥ limsup_{N→∞}\frac{1}{2}μ_{N}(|x^{(1)}| > K).

As the measures μ_{N} are finite the inequality μ_{N}(|x^{(1)}| > K) < ε holds for each index N with a constant K = K(N) that may depend on N. Hence the above inequality implies that formula (8.16) holds for all N ≥ 1 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 ε > 0 there exists some C(ε) < ∞ such that

μ_{N}( R^{l} \ [−C(ε),C(ε)]^{l})< ε for all N = 1,2,....

Consider the usual Fourier transforms

φ̃_{N}(t) = ∫_{R^{l}}e^{i(t,x)}μ_{N}(dx), t ∈ R^{l}.

Then

|φ_{N}(t) − φ̃_{N}(t)| ≤ 2ε + ∫_{[−C(ε),C(ε)]}|e^{i(t,x)}− e^{i([tC_{N}]/C_{N},x)} | μ_{N}(dx)
≤ 2ε + \frac{lC(ε)}{C_{N}}μ_{N}(R^{l})

for all ε > 0. Hence φ̃_{N}(t)−φ_{N}(t) → 0 as N → ∞, and φ̃_{N}(t) → φ(t). (Observe that μ_{N}(R^{l}) = φ_{N}(0) → φ(0) < ∞ as N → ∞, hence the measures μ_{N}(R^{l}) are uniformly bounded, and C_{N} → ∞ by the conditions of Lemma 8.4.) Then Lemma 8.4 follows from standard theorems on Fourier transforms. ⊓ ⊔

We return to the proof of Theorem 8.2. We apply Lemma 8.4 with C_{N} = N and l = kν for the measures μ_{N} defined in (8.15). Because of the middle term in (8.14) we can write the modified Fourier transform φ_{N} of the measure μ_{N} as

φ_{N}(t_{1},...,t_{k}) = ∑_{r=1}^{p} ∑_{s=1}^{p} c_{r}c_{s}ψ_{N}(t_{1}+ n_{r} − n_{s},...,t_{k}+ n_{r} − n_{s})

with

ψ_{N}(t_{1},...,t_{r}) = \frac{1}{N^{2ν}} ∫ exp\{ i\frac{1}{N}((j_{1},x_{1}) + ··· + (j_{k},x_{k}))\}

∑_{p∈B_{0}^{N}} ∑_{q∈B_{0}^{N}}exp\{ i( \frac{p − q}{N} ,x_{1}+ ··· + x_{k}) \} G_{N}(dx_{1})...G_{N}(dx_{k})

= \frac{1}{N^{2ν−kα} L(N)^{k}} ∑_{p∈B_{0}^{N}} ∑_{q∈B_{0}^{N}}r(p − q + j_{1})··· r(p − q + j_{k}),  (8.19)

where j_{p} = [t_{p}N], t_{p} ∈ R^{ν}, p = 1,...,k.

The asymptotical behaviour of ψ_{N}(t_{1},...,t_{k}) for N → ∞ can be investigated by the help of the last relation and formula (8.1). Rewriting the last double sum in the form of a single sum by fixing first the variable l = p−q ∈ [−N,N]^{ν} ∩ℤ_{ν}, and then summing up for l one gets

ψ_{N}(t_{1},...,t_{k}) = ∫_{[−1,1]^{ν}} f_{N}(t_{1},...,t_{k},x)dx

with

f_{N}(t_{1},...,t_{k},x)

= ( 1 − \frac{[|x^{(1)}N|]}{N}) ··· ( 1 − \frac{[|x^{(ν)}N|]}{N}) \frac{r([xN] + j_{1})}{N^{−α} L(N)} ···\frac{r([xN] + j_{k})}{N^{−α} L(N)}.

(In the above calculation we exploited that in the last sum of formula (8.19) the number of pairs (p,q) for which p − q = l = (l_{1},...,l_{ν}) equals (N − |l_{1}|)··· (N − |l_{ν}|).)

It can be seen with the help of formula (8.1) that for all ε > 0 the convergence

f_{N}(t_{1},...,t_{k},x) → f_{0}(t_{1},...,t_{k},x)  (8.20)

holds uniformly with the limit function

f_{0}(t_{1},...,t_{k},x) = (1 − |x^{(1)}|)...(1 − |x^{(ν)}|)\frac{a( \frac{x+t_{1}}{|x+t_{1}|})}{|x + t_{1}|^{α}} ... \frac{a( \frac{x+t_{k}}{|x+t_{k}|})}{|x + t_{k}|^{α}}

on the set x ∈ [−1,1]^{ν} \ ⋃_{p=1}^{k} {x: |x + t_{p}| > ε}.

We claim that

ψ_{N}(t_{1},...,t_{k}) → ψ_{0}(t_{1},...,t_{k}) = ∫_{[−1,1]^{ν}} f_{0}(t_{1},...,t_{k},x)dx,

and ψ_{0} is a continuous function.

This relation implies that μ_{N} →^{w} μ_{0}. To prove it, it is enough to show beside formula (8.20) that

|∫_{|x+t_{p}|<ε} f_{0}(t_{1},...,t_{k},x)dx|< C(ε), p = 1,...,k,  (8.21)

and

∫_{|x+t_{p}|<ε} |f_{N}(t_{1},...,t_{k},x)|dx < C(ε), p = 1,...,k, and N = 1,2,... (8.22)

with a constant C(ε) such that C(ε) → 0 as ε → 0.

By Hölder’s inequality

|∫_{|x+t_{p}|<ε} f_{0}(t_{1},...,t_{k},x)dx| ≤ C ∏_{1≤l≤k,l≠p} [∫_{x∈[−1,1]^{ν}} |x + t_{l}|^{−kα} dx]^{1/k}

[∫_{|x+t_{p}|≤ε} |x + t_{p}|^{−kα} dx]^{1/k} ≤ C′ε^{ν/k−α}

with some appropriate C > 0 and C′> 0, since ν − kα > 0, and a(·) is a bounded function. Similarly,

∫_{|x+t_{p}|<ε} |f_{N}(t_{1},...,t_{k},x)|dx ≤ ∏_{1≤l≤k,l≠p} [∫_{x∈[−1,1]^{ν}} \frac{|r([xN] + j_{l})|^{k}}{N^{−kα} L(N)^{k}} dx]^{1/k},

[∫_{|x+t_{p}|≤ε} \frac{|r([xN] + j_{p})|^{k}}{N^{−kα} L(N)^{k}} dx]^{1/k}.

It is not difficult to see, by using Karamata’s theorem, that if L(·) is a slowly varying function which is bounded in all finite intervals, then for all numbers η > 0 and K > 0 there is a threshold index N_{0} and a number C = C(N_{0},η,K) such that

L(uN) ≤ Cu^{−η} L(N) for all 0 < u < K and N ≥ N_{0}.

Hence formula (8.1) implies that

|r([xN] + j_{l})| = |r([xN] + [t_{l}N]) ≤ CN^{−α} L(N)(1 + |x + t_{l}|^{−α−η} ),  (8.23)

and

∫_{|x+t_{p}|<ε} \frac{|r([xN] + j_{p})|^{k}}{N^{−kα} L(N)^{k}} dx ≤ B∫_{|x+t_{p}|<ε} (1 + |x + t_{p}|^{−k(α+η)} )dx ≤ B′ε^{ν−k(α+η)}

∫_{x∈[−1,1]^{ν}} \frac{|r([xN] + j_{l})|^{k}}{N^{−kα} L(N)^{k}} dx ≤ B′′.

for a sufficiently small constant η > 0 with some constants B,B′,B′′< ∞ depending on η and t_{p}, 1 ≤ p ≤ k. (Let us remark that (8.23) holds also for |[xN]+j_{l}| ≤ K_{1} with some K_{1} > 0 independent of N, i.e. when the argument of r(·) is relatively small, because |r(n)| ≤ 1 for all n ∈ ℤ_{ν}.) Therefore we get, by choosing an η > 0 such that k(α + η) < ν, the inequality

∫_{|x+t_{p}|<ε} |f_{N}(t_{1},...,t_{k},x)|dx ≤ Cε^{ν/k−(α+η)}

with some C < ∞. The right-hand side of this inequality tends to zero as ε → 0. Hence we proved beside (8.20) formulae (8.21) and (8.22), therefore also the relation μ_{N} →^{w} μ_{0}. This means that with our choice of the functions K_{N}(·) and measures G_{N} all conditions of Lemma 8.3 are satisfied (if we know that also Lemma 8.1 holds), and its application yields Theorem 8.2. Thus we have proved Theorem 8.2 with the help of Lemma 8.1. ⊓ ⊔

It remains to prove Lemma 8.1.

Proof of Lemma 8.1. Introduce the notation

K_{N}(x) = ∏_{j=1}^{ν} \frac{e^{ix^{(j)}} − 1}{N(e^{ix^{(j)}/N}− 1)} , N = 1,2,...,

and

K_{0}(x) = ∏_{j=1}^{ν} \frac{e^{ix^{(j)}} − 1}{ix^{(j)}} .

Let us consider the measures μ_{N} defined in formula (8.15) in the special case k = 1, p = 1, c_{1} = 1. Then

μ_{N}(A) = ∫_{A}|K_{N}(x)|^{2}G_{N}(dx).

We have already seen in the proof of Theorem 8.2 that μ_{N} →^{w} μ_{0} with some finite measure μ_{0}, and the Fourier transform of μ_{0} is

φ_{0}(t) = ∫_{[−1,1]^{ν}} (1 − |x^{(1)}|)··· (1 − |x^{(ν)}|)\frac{a( \frac{x+t}{|x+t|})}{|x + t|^{α}} dx.

Moreover, since |K_{N}(x)|^{2} → |K_{0}(x)|^{2} uniformly in any bounded domain, it is natural to expect that G_{N} →^{v} G_{0} with G_{0}(dx) = \frac{1}{|K_{0}(x)|^{2}}μ_{0}(dx). But since K_{0}(x) = 0 in some points a direct proof of this statement would be difficult. Hence we choose a different approach to avoid these difficulties. First we prove a result about the vague convergence of the restrictions of the measures G_{N} to appropriate cubes.

We show that for all T ≥ 1 there is a finite measure G_{0}^{T} concentrated on (−T π,T π)^{ν} such that

lim_{N→∞} ∫ f(x)G_{N}(dx) = ∫ f(x)G_{0}^{T}(dx)  (8.24)

for all continuous functions f which vanish outside the cube (−T π,T π)^{ν}.

Let a continuous function f vanish outside the cube (−T π,T π)^{ν} with some T ≥ 1. Let M = [\frac{N}{2T}]. Then

∫ f(x)G_{N}(dx) = \frac{N^{α}}{L(N)}· \frac{L(M)}{M^{α}}∫ f ( \frac{N}{M}x) G_{M}(dx)

= \frac{N^{α} L(M)}{M^{α} L(N)}∫ f ( \frac{N}{M}x) |K_{M}(x)|^{−2} μ_{M}(dx)

→ (2T)^{α} ∫ f(2T x)|K_{0}(x)|^{−2} μ_{0}(dx)

= ∫ f(x)\frac{(2T)^{α}}{|K_{0}(\frac{x}{2T})|^{2}}μ_{0}( \frac{dx}{2T}) as N → ∞,

because f(\frac{N}{M}x)|K_{M}(x)|^{−2} vanishes outside the cube [−π,π]^{ν},

f(\frac{N}{M}x)|K_{M}(x)|^{−2} → f(2T x)|K_{0}(x)|^{−2} uniformly,

(the function K_{0}(·)^{−2} is continuous in the cube [−π,π]^{ν},) and μ_{M} →^{w} μ_{0} as N → ∞. Hence relation (8.24) holds. The measures G_{0}^{T} appearing in (8.24) are consistent for different parameters T, i.e. G_{0}^{T} is the restriction of the measure G_{0}^{T′}to the cube (−T π,T π)^{ν} if T′> T. This follows from the fact that ∫ f(x)G_{0}^{T}(dx) = ∫ f(x)G_{0}^{T′}(dx) for all continuous functions with support in (−T,T)^{ν}. It can be seen with the help of these facts that there is a locally finite measure G_{0} on R^{ν} such that G_{0}^{T} is its restriction to the cube (−T π,T π)^{ν}, and G_{N} →^{v} G_{0}.

Let us briefly explain why such a σ-finite measure G_{0} exists. The main problem is to show that the natural candidate for G_{0} is really a (σ-additive) measure. To show this let us represent the measures G_{0}^{T} as the Lebesgue–Stieltjes measures of appropriate functions Ḡ_{0}^{T}(x), x ∈ R^{ν}. To define these functions let us fix a number a = (a_{1},...,a_{ν}) such that the hyperplanes x_{j} = a_{j} have zero G_{0}^{T} measures for all 1 ≤ j ≤ ν and T. Then we can define the functions G_{0}^{T} as Ḡ_{0}^{T}(x_{1},...,x_{ν}) =

(−1)^{α(x)}G_{0}^{T}( ∏_{j=1}^{ν} [a_{j},x_{j})), where α(x) denotes for a vector x = (x_{1},...,x_{ν}) the

number of coordinates j, 1 ≤ j ≤ ν, such that x_{j} < a_{j}. Then the functions Ḡ_{0}^{T} have a limit function Ḡ_{0} as T → ∞, and G_{0} can be defined as the Lebesgue–Stieltjes measure defined by this function Ḡ_{0}.

As G_{N} →^{v} G_{0}, and |K_{N}(x)|^{2} → |K_{0}(x)|^{2} uniformly in all bounded regions, hence

μ_{N} →^{v} μ̄_{0}, where μ̄_{0}(A) = ∫_{A}|K_{0}(x)|^{2}G_{0}(dx), A ∈ B^{ν}. Since μ_{N} →^{w} μ_{0} the mea-

sures μ_{0} and μ̄_{0} must coincide, i.e.

μ_{0}(A) = ∫_{A}|K_{0}(x)|^{2}G_{0}(dx), A ∈ B^{ν}.

Relation (8.4) expresses the fact that φ_{0} is the Fourier transform of μ_{0}.

It remained to prove the homogeneity property (8.3) of the measure G_{0}. For this goal let us extend the definition of the measures G_{N} given in (8.2) to all non-negative real numbers u. It is easy to see that the relation G_{u} →^{v} G_{0} as u → ∞ remains valid. Hence we get for all fixed s > 0 and continuous functions f with compact support that

∫ f(x)G_{0}(dx)= lim_{u→∞} ∫ f(x)G_{u}(dx) = lim_{u→∞} \frac{s^{α} L(\frac{u}{s})}{L(u)}∫ f(sx)G_{\frac{u}{s}}(dx)
= s^{α} ∫ f(sx)G_{0}(dx) = ∫ f(x)s^{α} G_{0}( \frac{dx}{s}) .

This identity implies the homogeneity property (8.3) of G_{0}. Lemma 8.1 is proved. ⊓ ⊔

The next result is a generalization of Theorem 8.2.

Theorem 8.2′. Let X_{n}, n ∈ ℤ_{ν}, be a stationary Gaussian field with a correlation function r(n) defined in (8.1). Let H(x) be a real function with the properties EH(X_{n}) = 0 and EH(X_{n})^{2} < ∞. Let us consider the Fourier expansion

H(x) = ∑_{j=1}^{∞} c_{j}H_{j}(x), ∑ c_{j}^{2}j! < ∞,  (8.25)

of the function H(·) by the Hermite polynomials H_{j} (with leading coefficients 1). Let k be the smallest index in this expansion such that c_{k} ≠ 0. If 0 < kα < ν in (8.1), and the field Z_{n}^{N} is defined by the field ξ_{n} = H(X_{n}), n ∈ ℤ_{ν}, and formula (1.1), then the multi-dimensional distributions of the fields Z_{n}^{N} with A_{N} = N^{ν−kα/2} L(N)^{k/2} tend to those of the fields c_{k}Z_{n}^{∗}, n ∈ ℤ_{ν}, where the field Z_{n}^{∗} is the same as in Theorem 8.2.

Proof of Theorem 8.2′. Define H′(x) = ∑_{j=k+1}^{∞} c_{j}H_{j}(x) and Y_{n}^{N} = \frac{1}{A_{N}} ∑_{l∈B_{n}^{N}}H′(X_{l}). Be-

cause of Theorem 8.2 in order to prove Theorem 8.2′it is enough to show that

E(Y_{n}^{N})^{2} → 0 as N → ∞.

It follows from Corollary 5.5 that EH_{j}(X_{n})H_{l}(X_{m}) = δ_{j,l}j!(EX_{n}X_{m})^{j} = δ_{j,l}j!r(n − m)^{j}, where δ_{j,l} = 0 if j ≠ l, and δ_{j,l} = 1 if j = l. Hence

E(Y_{n}^{N})^{2} = \frac{1}{A_{N}^{2}} ∑_{j=k+1}^{∞} c_{j}^{2}j! ∑_{s,t∈B_{n}^{N}}[r(s − t)]^{j}.

Some calculation yields with the help of this identity and formula (8.1) that

E(Y_{n}^{N})^{2} = \frac{1}{A_{N}^{2}}[O(N^{2ν−(k+1)α} L(N)^{k+1}) + O(N^{ν})]→ 0.

Theorem 8.2′is proved. ⊓ ⊔

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 ∈ ℤ_{ν} with a correlation function satisfying relation (8.1), and the field ξ_{n}= H(X_{n}), n ∈ ℤ_{ν}, subordinated to it with a general function H(x) such that EH(X_{n}) = 0 and EH(X_{n})^{2} < ∞. We are interested in the large-scale limit of such random fields. Take the Hermite expansion (8.25) of the function H(x), and let k be the smallest such index for which c_{k} ≠ 0 in the expansion (8.25). In Theorem 8.2′we solved this problem if 0 < kα < ν. We are interested in the question what happens in the case when kα > ν. Let me remark that in the case kα ≥ ν the field Z_{n}^{∗}, n ∈ ℤ_{ν}, which appeared in the limit in Theorem 8.2′does not exist. The Wiener-Itô integral defining Z_{n}^{∗} is meaningless, because the integral which should be finite to guarantee the existence of the Wiener–Itô integral is divergent in this case. Next I formulate a general result which contains the answer to the above question as a special case.

Theorem 8.5. Let us consider a stationary Gaussian random field X_{n}, EX_{n} = 0, EX_{n}^{2} = 1, n ∈ ℤ_{n}, with correlation function r(n) = EX_{m}X_{m+n}, m,n ∈ ℤ_{ν}. Take a function H(x) on the real line such that EH(X_{n}) = 0 and EH(X_{n})^{2} < ∞. Take the Hermite expansion (8.25) of the function H(x), and let k be smallest index in this expansion such that c_{k} ≠ 0. If

∑_{n∈ℤ_{ν}} |r(n)|^{k} < ∞,  (8.26)

then the limit

lim_{N→∞} EZ_{n}^{N}(H_{l})^{2} = lim_{N→∞} N^{−ν} ∑_{i∈B_{n}^{N}} ∑_{j∈B_{n}^{N}} r^{l}(i − j) = σ_{l}^{2}l!

exists for all indices l ≥ k, where Z_{n}^{N}(H_{l}) is defined in (1.1) with A_{N}= N^{ν/2}, and ξ_{n}= H_{l}(X_{n}) with the l-th Hermite polynomial H_{l}(x) with leading coefficient 1. Moreover, also the inequality

σ^{2} = ∑_{l=k}^{∞} c_{l}^{2}l!σ_{l}^{2} < ∞

holds.

The finite dimensional distributions of the random field Z_{n}^{N}(H) defined in (1.1) with A_{N} = N^{ν/2} and ξ_{n} = H(X_{n}) tend to the finite dimensional distributions of a random field σZ_{n}^{∗} with the number σ defined in the previous relation, where Z_{n}^{∗}, n ∈ ℤ_{ν}, are independent, standard normal random variables.

Theorem 8.5 can be applied if the conditions of Theorem 8.2′ hold with the only modification that the condition kα < ν is replaced by the relation kα > ν. In this case the relation (8.26) holds, and the large-scale limit of the random field Z_{n}^{N}, n ∈ ℤ_{ν} with normalization A_{N} = N^{ν/2} is a random field consisting of independent standard normal random variables multiplied with the number σ. There is a slight generalization of Theorem 8.5 which also covers the case kα = ν. In this result we assume instead of the condition (8.26) that ∑_{n∈B̄_{N}}r(n)^{k} = L(N) with a slowly

varying function L(·), where B̄_{N}= {(n_{1},...,n_{ν}) ∈ ℤ_{ν}: −N ≤ n_{j} ≤ N, 1 ≤ j ≤ ν}, and some additional condition is imposed which states that an appropriately defined finite number σ^{2} = lim_{N→∞} σ_{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^{ν/2}L(N)^{1/2}. This result has the consequence that if the conditions of Theorem 8.2′ hold with the only difference that kα = ν instead of kα < ν, then the large scale limit exists with norming constants A_{N} = N^{ν/2}L(N) with an appropriate slowly varying function L(·), 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 [3]. 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 → ∞. 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–Itô integral or apply a version of the diagram formula which gives the moments of Wick polynomials instead of Wiener– Itô 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 [3] 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}},…, {k_{M−1},k_{M}} of the set {1,...,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 ≤ j ≤ \frac{M}{2}, 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.

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

ξ_{n} = ∑_{j=k}^{∞} \frac{1}{j!} ∫ e^{i(n,x_{1}+··· +x_{j})} α_{j}(x_{1},...,x_{j})Z_{G}(dx_{1})...Z_{G}(dx_{j}), n ∈ ℤ_{ν}, (8.27)

where Z_{G} is the random spectral measure adapted to a Gaussian field X_{n}, n ∈ ℤ_{ν}, with correlation function satisfying (8.1) with 0 < α < \frac{ν}{k}.

Theorem 8.6. Let the fields Z_{n}^{N} be defined by formulae (8.27) and (8.1) with A_{N} = N^{ν−kα/2} . The multi-dimensional distributions of the fields Z_{n}^{N} tend to those of the field α_{k}(0,...,0)Z_{n}^{∗} where the field Z_{n}^{∗} is the same as in Theorem 8.2 if the following conditions are fulfilled:

(i) α_{k}(x_{1},...,x_{k}) is a bounded function, continuous at the origin, and such that α_{k}(0,...,0) ≠ 0.

(ii)

∑_{j=k=1}^{∞} \frac{1}{j!} \frac{N^{−(j−k)α}}{L(N)^{j−k}} ∫_{R^{jν}} |α_{j} ( \frac{x_{1}}{N},..., \frac{x_{j}}{N}) |^{2} \frac{1}{N^{2ν}} | ∑_{j∈B_{0}^{N}} e^{i(l/N,x_{1}+··· +x_{j})} |^{2}
G_{N}(dx_{1})...G_{N}(dx_{j}) → 0,

where G_{N} is defined in (8.2).

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) ξ_{n} can be substituted in the present proof by the following expression:

ξ′_{n}= \frac{1}{k!} ∫ e^{i(n,x_{1}+··· +x_{k})} α_{k}(x_{1},...,x_{k})Z_{G}(dx_{1})...Z_{G}(dx_{k}), n ∈ ℤ_{ν}.

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 μ_{N} defined in formula (8.15) by the following measure μ̄_{N}:

μ̄_{N}(A) = ∫_{A}|K_{N}(x_{1},...,x_{k})|^{2} |α_{k}( \frac{x_{1}}{N},..., \frac{x_{k}}{N}) |^{2}G_{N}(dx_{1})...G_{N}(dx_{k}),
A ∈ B^{kν},

and to observe that because of condition (i) the limit relation μ_{N} →^{w} μ_{0} implies that

μ̄_{N} →^{w} |α_{k}(0,...,0)|^{2}μ_{0}. ⊓ ⊔

The main problem in applying Theorem 8.6 is to check conditions (i) and (ii). We remark without proof that any field ξ_{n} = H(X_{s_{1}+n} ,...,X_{s_{p}+n}), s_{1},...,s_{p} ∈ ℤ_{ν} and n ∈ ℤ_{ν}, for which Eξ_{n}^{2} < ∞ satisfies condition (ii). This is proved in Remark 6.2 of [9]. 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 [24] for a proof.) Here we restrict ourselves to the case ν = 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 = ...,−1,0,1,..., be a sequence of real numbers such that

\begin{matrix}a_{n} = C(1)n^{−β−1} + o(n^{−β−1} ) if n ≥ 0 \\ a_{n} = C(2)|n|^{−β−1} + o(|n|^{−β−1} ) if n < 0 \end{matrix}  − 1 < β < 1.  (8.28)

Let X_{n}, n = ...,−1,0,1,..., be a stationary Gaussian sequence with correlation function r(n) = EX_{0}X_{n} = |n|^{−α} L(|n|), 0 < α < 1, where L(·) is a slowly varying function. Define the field ξ_{n}, n = ...,−1,0,1,..., as

ξ_{n} = ∑_{m=−∞}^{∞} a_{m}H_{k}(X_{m+n}).  (8.29)

Theorem 8.7. Let a sequence ξ_{n}, n = ...,−1,0,1,..., be defined by (8.28) and (8.29). Let 0 < kα < 1, 0 < 1−β − \frac{k}{2}α < 1, and let one of the following conditions be satisfied.

(a) 0 < β < 1, and ∑_{n=−∞}^{∞} a_{n} = 0.

(b) 0 > β > −1.

(c) β = 0, C(1) = −C(2), and ∑_{n=0}^{∞} |a_{n}+ a_{−n} | < ∞.

Let us define the sequences Z_{n}^{N} by formula (1.1) with A_{N} = N^{1−β−kα/2} L(N)^{k/2} and the above defined field ξ_{n}. The multi-dimensional distributions of the sequences Z_{n}^{N} tend to those of the sequences D^{−k} Z_{n}^{∗}(α,β,a,b,c), where

Z_{n}^{∗}(α,β,k,b,c) = ∫ χ̃_{n}(x_{1}+ ··· + x_{k})
[b|x_{1}+ ··· + x_{k}|^{β} + ic|x_{1}+ ··· + x_{k}|^{β}sign(x_{1}+ ··· + x_{k})]
|x_{1}|^{(α−1)/2} ··· |x_{k}|^{(α−1)/2} W (dx_{1})...W (dx_{k}),

W (·) 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 = 2Γ (α)cos(\frac{α}{2}π), and

b = 2[C(1)+C(2)]Γ (−β)sin(\frac{β+1}{2} π), and c = 2[C(1)−C(2)]Γ (−β)cos(\frac{β+1}{2} π) in cases (a) and (b), and b = ∑_{n=−∞}^{∞} a_{n}, and c = C(1) in case (c).

History of the problems. Comments

Chapter 1.

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 [33]. (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 original 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 [28]. Further results in this direction were proved by R. L. Dobrushin, H. Kesten and F. Spitzer, P. Major, M. Rosenblatt and M. S. Taqqu [7], [8], [9], [24], [29], [30], [34], [37]. In most of these papers only the one-dimensional case is considered, i.e. the case when R^{ν} = R^{1}, and it is formulated in a different but equivalent way. The joint distribution

of the random variables A_{N}^{−1} ∑_{j=1}^{Nt]}ξ_{j}, 0 < t < ∞, is 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 [2], [6], [17], [23]. It was proved that in these models the limit is, with a non-typical normalization, a Gaussian selfsimilar field. 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 [7]. 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 [1] who can define the required formal power series. 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–Itô integrals contains the non-Gaussian selfsimilar fields interesting for statistical physics. Some experts are very skeptical in this respect. The Gaussian self-similar fields are investigated in [7] and [32]. A more thorough investigation is under preparation in [11].

The notion of generalized fields was introduced by I. M. Gelfand. A detailed discussion can be found in the book [15], where the properties of Schwartz spaces we need can also be found.

In the definition of generalized fields the class of test functions S can be substituted by other linear topological spaces consisting of real valued functions. The most frequently considered space, beside the space S, is the space D of infinitely many times differentiable functions with compact support. In paper [7] Dobrushin also considered the space S^{r} ⊂ S, which consists of the functions φ ∈ S satisfying the additional relation ∫ x^{(1)}^{j_{1}} ··· x^{(ν)}^{j_{ν}} φ(x)dx = 0, provided that j_{1}+ ··· + j_{ν} < r. He considered this class of test functions, because there are much more continuous linear functionals over S^{r} than over S, and this property of S^{r} can be exploited in certain investigations. Generally no problem arises in the proofs if the space of test functions S is substituted by S^{r} or D in the definition of generalized fields.

Two generalized fields X(φ) and X̄(φ) can be identified if X(φ) =^{Δ} X̄(φ) for all φ ∈ S. Let me remark that this relation also implies that the multi-dimensional distributions of the random vectors (X(φ_{1}),...,X(φ_{n})) and (X̄(φ_{1}),...,X̄(φ_{n})) coincide for all φ_{1},...,φ_{n} ∈ S. As 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 S′denote the space of continuous linear functionals over S, and let A_{S′} be the σ-algebra over S′generated by the sets A(φ,a) = {F : F ∈ S′;, F (φ) < a}, where φ ∈ S and a ∈ R^{1} are arbitrary. Given a probability space (S′,A_{S′},P), a generalized field X̄ = X̄(φ) can be defined on it by the formula X̄(φ)(F) = F (φ), φ ∈ S, and F ∈ S′. The following deep result is due to Minlos (see e.g. [15]).

Theorem. (Minlos.) Let (X(φ), φ ∈ S) be a generalized random field. There exists a probability measure P on the measurable space (S ,A_{S′}) such the generalized field X̄= (X̄(φ), φ ∈ S) defined on the probability space (S̄,A_{S′},P) by the formula X̄(φ)(F) = F (φ), φ ∈ S, F ∈ S′, satisfies the relation X(φ) =^{Δ} X̄(φ) for all φ ∈ S.

The generalized field X̄has some nice properties. Namely property (a) in the definition of generalized fields holds for all F ∈ S′. Moreover X̄satisfies the following strengthened version of property (b):

(b′) limX̄(φ_{n}) = X̄(φ) in every point F ∈ S′if φ_{n} → φ in the topology of S.

Because of this nice behaviour of the field X̄(φ) most authors define generalized fields as the versions X̄ defined in Minlos’ theorem. Since we have never needed the extra properties of the field X̄ 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 S, namely that S is a so-called nuclear space. Minlos’ theorem also holds if the space of test functions is substituted by D or S^{r} in the definition of generalized fields.

Let us finally remark that Lamperti [21] gave an interesting characterization of self-similar random fields. Let X(t), t ∈ R^{1}, be a continuous time stationary random process, and define the random process Y (t) = \frac{X(logt)}{t^{α}}, t > 0, with some α > 0. Then, as it is not difficult to see, the random processes Y (t), t > 0, and \frac{Y (ut)}{u^{α}}, 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 α > 0 on the half-line t > 0. Contrariwise, if the finite dimensional distributions of the processes Y (t) and \frac{Y (ut)}{u^{α}}, t > 0, agree for all u > 0, then the process X(t) = \frac{X(e^{t})}{e^{αt}}, t ∈ 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.

Chapter 2.

Wick polynomials are widely used in the literature of statistical physics. A detailed discussion about Wick polynomials can be found in [12]. 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 [13] (Lemma 1.5). Theorem 2.1 is due to Segal [31]. It is closely related to a result of Cameron and Martin [4]. The remarks at the end of the chapter about the content of formula 2.1 are related to [25].

Chapter 3.

Random spectral measures were independently introduced by Cramer and Kolmogorov [5], [20]. 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 [15]. Let me remark that the same result holds true if the space of test functions S is substituted by D.

Chapter 4.

The stochastic integral defined in this chapter is a version of that introduced by Itô in [18]. This modified integral first appeared in Totoki’s lecture note [38] in a special form. Its definition is a little bit more difficult than the definition of the original stochastic integral introduced by Itô, 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 [7]. The definition of Wiener–Itô 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–Itô integrals can be well approximated by simple functions.

Chapter 5.

Proposition 5.1 is proved for the original Wiener–Itô integrals by Itô in [18]. 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 [7]. 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 [35] 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 moment of products of Wick polynomials and not only of Hermite polynomials.

I could not find results similar to Propositions 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 [27], which I formulate below.

Let X_{t}, t ∈ T, and Y_{t′}, t′∈ T′be two sets of jointly Gaussian random variables on some probability spaces (Ω,A ,P) and (Ω,A′,P′). Let H_{1} and H′_{1}be the Hilbert spaces generated by the finite linear combinations ∑ c_{j}X_{t_{j}} and ∑ c_{j}Y_{t′_{j}}. Let us define the σ-algebras B = σ(X_{t}, t ∈ T) and B′= σ(Y_{t′}, t′∈ T′) and the Banach spaces L_{p}(X) = L_{p}(Ω,B,P), L_{p}(Y) = L_{p}(Ω′,B′,P′), 1 ≤ p ≤ ∞. Let A be linear transformation from H_{1} to H′_{1}with norm not exceeding 1. We define an operator Γ (A): L_{p}(X) → L_{p′}(Y) for all 1 ≤ p,p′≤ ∞ in the following way. If η is a homogeneous polynomial of the variables X_{t},

η = ∑ C_{j_{1},...,j_{s}}^{t_{1},...,t_{s}}X_{t_{1}}^{j_{1}} ··· X_{t_{s}}^{j_{s}}, t_{1},...,t_{s} ∈ T,

then

Γ (A): η: = ∑ C_{j_{1},...,j_{s}}^{t_{1},...,t_{s}}: (AX_{t_{1}})^{j_{1}} ··· (AX_{t_{s}})^{j_{s}}: .

It can be proved that this definition is meaningful, i.e. Γ (A): η: does not depend on the representation of η, and Γ (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 Γ (A)ξ is defined for all ξ ∈ L_{p}(X), p ≥ 1. Nelson’s hypercontractive inequality says the following. Let A be a contraction from H_{1} to H′_{1}. Then Γ (A) is a contraction from L_{q}(X) to L_{p}(Y) for 1 ≤ q ≤ p provided that

∥A∥ ≤ ( \frac{q − 1}{p − 1} ) ^{1/2}.  (9.1)

If (9.1) does not hold, then Γ (A) is not a bounded operator from L_{q}(X) to L_{p}(Y). A further generalization of this result can be found in [16]. 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 ∈ T) = (Y_{t′}, t′∈ T′) is a stationary Gaussian field with spectral measure G, q = 2, p = 2m with some positive integer m, A = c · Id, where Id denotes the identity operator, and c = (2m − 1)^{−1/2} . Let H ^{c} and H_{n}^{c} be the complexification of the real Hilbert spaces H and H_{n} defined in Chapter 2. Then L_{2}(X) = H ^{c} = H_{0}^{c} + H_{1}^{c} + ···by Theorem 2.1 and formula 2.1. The operator Γ (c · Id) equals c^{n}· Id on the subspace H_{n}^{2}. If h_{n} ∈ H_{G}^{n}, then I_{G}(h_{n}) ∈ H_{n}, hence the application of Nelson’s inequality for the operator A = c · Id shows that

(EI_{G}(h_{n})^{2m})^{1/2m}= c^{−n} (E(Γ (c · Id)I_{G}(h_{n}))^{2m})^{1/2m}≤ c^{−n} (EI_{G}(h_{n})^{2})^{1/2}

i.e.

EI_{G}(h_{n})^{2m}≤ c^{−2nm} (EI_{G}(h_{n})^{2})^{m}= (2m − 1)^{mn}(EI_{G}(h_{n})^{2})^{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: H_{1} → H′_{1} be a contraction ∥A∥ = c < 1. Then there exists a p̄= p̄(c) > 2 such that Γ (A) is a bounded operator from L_{2}(X) to L_{p}(Y) for p < p̄. This weakened form of Nelson’s inequality is sufficient in many applications.

Chapter 6.

Theorems 6.1, 6.2 and Corollary 6.4 were proved by Dobrushin in [7]. Taqqu proved similar results in [36], but he gave a different representation. Theorem 6.6 was proved by H. P. Mc.Kean in [26]. The proof of the lower bound uses some ideas from [14]. Remark 6.5 is from [24]. As Proposition 6.3 also indicates, some nontrivial 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. [22]) was worked out. This method could also be applied in this chapter. Part (b) of Proposition 6.3 implies that the self-similarity parameter α cannot be chosen in a larger domain in Corollary 6.4. One can ask about the behaviour of the random variables ξ_{j} and ξ(φ) defined in Corollary 6.4 if the self-similarity parameter α tends to the critical value \frac{ν}{2}. The variance of the random variables ξ_{j} and ξ(φ) tends to infinity in this case, and the fields ξ_{j}, j ∈ ℤ_{ν}, and ξ(φ), φ ∈ 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 will appear in [10].

In a recent paper [19] 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 selfsimilar 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), −∞ < t < ∞, be two independent Wiener processes. (We say that B(t) is a Wiener process on the real line if B(t), t ≥ 0, and B(−t), t ≥ 0, are two independent Wiener processes.) Let K(x,t_{1},t_{2}), x ∈ 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} = ∫ K(x,n,n + 1)B_{2}(dx), n = ...,−1,0,1,...,

where the integral in the last formula is an Itô integral, is a stationary self-similar field with self-similarity parameter \frac{3}{4}.

To see the self-similarity property one has to observe that

K(λ^{1/2}x,λt_{1},λt_{2}) =^{Δ} λ^{1/2}K(x,t_{1},t_{2}) for all x ∈ R^{1}, t_{1} < t_{2}, and λ > 0

because of the relation B_{1}(λu) =^{Δ} λ^{1/2}B_{1}(u). Hence

∑_{j=0}^{n−1} Z_{j} =^{Δ} n^{1/2}∫ K(n^{−1/2} x,0,1)B_{2}(dx) =^{Δ} n^{3/4}∫ 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 (1.1) can be seen similarly.

To see the stationarity of the field Z_{n} we need the following two observations.

(a) K(x,s,t) =^{Δ} K(x + η(s),0,t − s) with η(s) = −B_{1}(−s). (The form of η is not important for us. What we need is that the pair (η,K) is independent of B_{2}.)

(b) If α(x), −∞ < x < ∞, is a process independent of B_{2}, then

∫ α(x + u)B_{2}(dx) =^{Δ} ∫ α(x)B_{2}(dx) for all u ∈ R^{1}.

It is enough to show, because of Property (a) that

∫ K(x + η(s),0,t − s)B_{2}(dx) =^{Δ} ∫ 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 η(s) = u, u ∈ R^{1}.

The generalized field version of the above field Z_{n} is the field

Z(φ) = −∫ [K(x,0,t)\frac{dφ}{dt} dt]B_{2}(dx), φ ∈ S .

To explain the analogy between the field Z_{n} and Z(φ) we remark that the kernel of the integral defining Z_{n} can be written, at least formally, as

K(x,n,n + 1) = ∫ χ_{[n,n+1)}(u)\frac{d}{du}K(x,n,u)du,

although K is a non-differentiable function. Substituting the function χ_{[n,n+1)} by φ ∈ 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(φ).

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.

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–Itô integrals with respect to a white noise field. Then the process Z_{n} itself can also be represented via multiple Wiener–Itô 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–Itô integrals with the help of a randomization.

Chapter 7.

The definition of Wiener–Itô integrals together with the proof of Theorem 7.1 and Proposition 7.3 are given by Itô in [18]. Theorem 7.2 is proved in Taqqu’s paper [37]. He needed this result to show that the self-similar fields defined in [9] by means of Wiener–Itô integrals coincide with the self-similar fields defined in [37] by means of modified Wiener–Itô integrals.

Chapter 8.

The results of this chapter, with the exception of Theorems 8.5 and 8.7 are proved in [9]. Theorem 8.5 is proved in [3] and Theorem 8.7 in [24]. The latter paper was strongly motivated by [29]. Lemma 8.3 is formulated in a more general form than Lemma 3 in [9]. 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^{kν}, 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}+··· +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 [9].

The example for non-central limit theorems given by Rosenblatt in [28] and its generalization by Taqqu in [34] are special cases of Theorem 8.2. In these papers only the special case H_{2}(x) = x^{2}− 1 is considered. Later Taqqu [37] 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 [34].

The method of [28] and [34] does not apply for the proof of Theorem 8.2 in the case of H_{k}(x), k ≥ 3. 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} →^{D} 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–Itô integral behaves similarly to the 2kn-the moment of a Gaussian random variable with zero expectation, it equals e^{(knlogn)/2+O(n)}. Hence some results about the so-called moment problem show that the distribution of a k-fold Wiener–Itô 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 ≥ 3.

Throughout Chapter 8 I have assumed that the correlation function of the underlying Gaussian field to which our fields are subordinated satisfies formula (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 [9]. 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 [9] 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 [30] 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 selfsimilar fields defined in formula (6.6) have a much greater range of attraction if the homogeneous function f_{n} in (6.6) 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 [24].

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Index

B

Bochner theorem 15, 22 Bochner–Schwartz theorem 15, 23

C

canonical representation of a subordinated

field 65 complete diagram 56 convergence of generalized random fields in

distribution 4

D

diagram 45 diagram formula 46 discrete random field 1

F

Fock space 28, 78 formula for the change of variables in

Wiener–Itô integrals 37 Fourier transform of a generalized function

7

G

Gaussian random field (generalized field) 4 Gaussian random orthogonal measure 75 Gaussian random spectral measure. 18, 19 generalized functions with the test function

space of smooth functions with compact support 8

generalized random field 4 H

Hermite polynomials 9

I

Itô’s formula 36

Itô’s formula for Wiener–Itô integrals with respect to a random orthogonal measure 77

K

Karamata’s theorem 81

M

Minlos’s theorem 106

modified Fourier transform 92

multiplicatively positive generalized function 23

N

Nelson’s hypercontractive inequality 108

O

one-fold stochastic integral 18

P

positive definite function 21

positive definite generalized function 23 positive definite Hermitian bilinear functional

24 positive generalized function 22 Prokhorov metric in the space of probability

measures 87 R

random field (generalized) subordinated to a generalized stationary Gaussian random field 5, 62

random field subordinated to a stationary Gaussian random field (discrete case) 2, 61

random spectral measure adapted to a Gaussian

random field 19 regular system of sets in R^{ν} 29

S

Schwartz space of generalized functions. 6 self-similar (discrete) random field 2 self-similar random field (generalized field)

4 self-similarity parameter 2, 4 shift transformation (discrete random field)

2, 10 shift transformation (generalized random field) simple function 29, 77 slowly varying function 81 space of test functions in the Schwartz space

of generalized functions 3, 6 spectral measure of a stationary random field

(for discrete and generalized random fields) 16

stationary (discrete) random fields 1 stationary random field (generalized field) 4

V

vague convergence of measures 82

W

weak convergence of probability measures

87, 91 white noise 78 Wick polynomials 11 Wiener–Itô integral 36 Wiener–Itô integrals with respect to a random

orthogonal measure 76

Acronyms

D The space of infinitely differentiable functions with compact support D′ The space of generalized functions on the space of test function D d(A) The diameter of the set A ExpH_{G} and ExpK_{μ} The Fock space G(·) The spectral measure of a stationary discrete or generalized random field G_{n} →^{v} G_{0} The vague convergence of the locally finite measures G_{n} to the locally

finite measure G_{0} H_{n}(x) The Hermite polynomial of order n with leading coefficient 1 H The (real) Hilbert space of square-integrable random variables measurable

with respect to the σ-algebra generated by the random variables of a previously defined Gaussian random field

H_{1} The smallest subspace of the Hilbert space H containing the elements of

underlying Gaussian field H_{≤n} The smallest subspace of the Hilbert space H containing the polyomials of

order less than or equal to n of the random variables in the underlying Gaussian field

H_{n} The orthogonal completion of the subspace H_{≤n−1} in the Hilbert space H_{≤n} h_{γ}(·) The kernel function of the Wiener–Itô integral appearing in the diagram

formula and depending on the diagram γ H̄_{G}^{n} The space of functions which can be the kernel function of an n-fold Wiener–

Itô integral with respect to a random spectral measure Z_{G}with spectral measure G H_{G}^{n} The subspace of H̄_{G}^{n} consisting of symmetric functions ˆ¯H _{G}^{n} The subspace of H̄_{G}^{n} consisting of simple functions Ĥ_{G}^{n} The subspace of H_{G}^{n} consisting of simple (and symmetric) functions I_{G}(f_{n}) The normalized Wiener–Itô integral of the kernel function f_{n} of n variables

with respect to the random spectral measure Z_{G} K̄_{μ}^{n} The class of function which can be the kernel function of an n-fold Wiener–

Itô integral with respect to a random orthogonal measure Z_{μ} K_{μ}^{n} The subspace of K̄_{μ}^{n} consisting of symmetric functions ˆ¯K_{μ}^{n} The set of simple functions appearing in the definition of n-fold Wiener–Itô

integrals with respect to a random orthogonal measure Z_{μ} K The Hilbert space of square integrable random variables measurable with re-

spect to the σ-algebra generated by the random variables Z_{μ}(A) of a random orthogonal measure Z_{μ}

K_{≤n} The subspace of K generated by the polynomials of the random variables Z_{μ}(A) of the orthogonal random field Z_{μ} which have order less than or equal to n

K_{n} The orthogonal completion of the subspace K_{≤n−1} in the Hilbert space K_{≤n} : P(ξ_{1},...,ξ_{n}): The Wick polynomial corresponding to the polynomial

P(x_{1},...,x_{n}) and Gaussian random vector (ξ_{1},...,ξ_{n}) S The class of test functions in the Schwartz space S^{c} The class of complex number valued test functions in the Schwartz space S′ The Schwartz space of generalized functions S_{ν−1} The ν-dimensional unit sphere Symf The symmetrization of the function f T_{m} and T_{t} The shift operator with parameter m ∈ ℤ_{ν} and t ∈ R^{ν} X(φ) The value of the generalized field X(·) at the test function φ Z_{G}(·) The (Gaussian) random spectral measure corresponding to the spectral

measure G Z(dx) The (Gaussian) random spectral measure whose spectral measure is \frac{1}{2π}

times the Lebesgue measure on [−π,π) Z_{μ}(·) The random orthogonal measure corresponding to the measure μ ℤ_{ν} The set of lattice points in the ν-dimensional space with integer coordinates Γ (n_{1},...,n_{m}) The space of diagrams in the diagram formula Γ̄ The space of closed diagrams |γ| The number of edges in a diagram γ μ_{n} →^{w} μ The weak convergence of the probability measures μ_{n} to the probability

measure μ ξ_{N} →^{D} ξ_{0} The convergence of the random variables ξ_{N} to the random variable ξ_{0}

in distribution, i.e. the weak convergence of the distributions of ξ_{N} to the distribution of ξ_{0}

Π_{n} The group of permutations of the set {1,...,n} ρ_{p}(μ,ν) The Prokhorov metric of the probability measures μ and ν χ_{A}(·) The indicator function of the set A. χ̃_{n}(x) The Fourier transform of the indicator function of the unit cube

∏_{p=1}^{ν} [n^{(p)},n^{(p)} + 1), where n = (n^{(1)},...,n^{(p)})

⊖ The orthogonal completion of a subspace of a Hilbert space ˜ Fourier transform

∗ Convolutions =^{Δ} Identity in distribution ⇒ Stochastic convergence ∫′ Wiener–Itô integral with respect to a random orthogonal measure

[x] Integer part of a real number x Abstract: Chapter 1

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 its 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.

Chapter 2

We introduce the notion of Wick polynomials which are natural multi-variate versions of Hermite polynomials. We present their most important properties, and we give with their help 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.

Chapter 3

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.

Chapter 4

Here we introduce the multiple Wiener–itô integrals with respect to a Gaussian random spectral measure and prove some important results about them.

Chapter 5

Here we prove the most important result about multiple Wiener–Itô integrals, the so-called diagram formula together with some of its consequences. In the diagram formula we rewrite the product of Wiener–Itô integrals in the form of a sum of Wiener–Itô 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.

Chapter 6

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.

Chapter 7.

Here we discuss the original Wiener–Itô 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–Itô integrals with respect to a random spectral measure and to the classical Itô integrals of a stochastic process.

Chapter 8.

Here we present some non-trivial limit theorem 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.