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\topmatter
\title Asymptotic distributions for weighted $\bold
U$-statistics\endtitle \author P\'eter Major\endauthor
 
\rightheadtext{Weighted $\bold U$-statistics}
 
\affil Mathematical Institute of the Hungarian Academy of Sciences
\endaffil
\abstract We prove limit theorems for weighted $U$-statistics and
express the limit by means of multiple stochastic integrals. This is a
generalization of the paper of K.~A.~O'Neil and R.~A. Redner ~[9].
In that paper the method of moments was applied which does not work
in the general case. Hence we had to work out a diffferent method.
In particular,  in Theorem ~4 we describe the limit of a model proposed
by O'Neil and Redner. In this model the weight functions cause an
intricate cancelletion, and the limit can be presented as a sum of
multiple stochastic integrals with different multiplicities. \endabstract
\endtopmatter
\subheading{1. Introduction} In this paper we investigate the limit
behavior of
weighted $U$-statistics which means statistics of the following
form:
$$
U_n=\sum_{1\le j_1<j_2\dots<j_k\le n}
a(j_1,\dots,j_k)f(X_{j_1},\dots,X_{j_k})\;. \tag $1.1$
$$
Here $X_1$, \dots, $X_n$ are iid.\ random
variables with uniform distribution in the interval $[0,1]$, the
functions $a(x_1,\dots, x_k)$ and $f(x_1,\dots, x_k)$ are symmetric,
i.e.\ they are invariant under all permutations of their arguments,
and the function $f$ also satisfies the condition
$$
\int_{[0,1]^k} f^2(x_1,\dots,x_k)\,dx_1\dots\,dx_k<\infty\;.\tag1.2
$$
The expression (1.1) is a generalization of the usual (unweighted)
$U$-statistics, investigated e.g.\
in [1], because of the appeareance of a weight function $a(x_1,\dots,
x_k)$ in it. The assumption that the sequence of iid.\ random
variables $X_1,\dots,X_n$ is uniformly distributed is not a real
restriction. If its distribution function is $F(x)$, then the sequence
$F(X_1),\dots,F(X_n)$ is uniformly distributed, and the statistics $U_n$
do not change if the function $f(x_1,\dots,x_k)$ is replaced by
$f(F^{-1}(x_1),\dots,F^{-1}(x_k))$  and the random variables $X_j$,
by $F(X_j)$, $j=1,\dots,n$.  In most results of this paper we restrict our
attention
to the so-called degenerate $U$-statistics, i.e.\ we assume that
$$
\int f(y, x_2,\dots,x_k)\,dy=0 \quad\text{for all }x_2,\dots,x_k\;.
\tag1.3
$$
The investigation of $U$-statistics with general
kernel functions $f$ can be reduced to this special case by means of the
Hoeffding decomposition. (See e.g.\ Appendix ~A in~[1].) This gives the
following representation of a symmetric function $f(x_1,\dots,x_k)$ with
$k$ arguments: There exists a (unique)
sequence  of symmetric functions $f_s=f_s(x_1,\dots,x_s)$, \
$s=1,\dots,k$, and a constant $f_0$ such that
$$
f(x_1,\dots,x_k)=f_0+\sum_{s=1}^k\sum_{\{i_1,\dots,i_s\}
\subset\{1,\dots,k\}}f_s(x_{i_1},\dots,x_{i_s})\;,\tag1.4
$$
and the functions $f_s$ are degenerate, i.e.
$$
\int f_s(y, x_2,\dots,x_s)\,dy=0 \quad\text{for all }x_2,\dots,x_s,\quad
2\le s\le k\;.\tag$1.4'$
$$
Because of this decomposition the
investigation of the limit behavior of the statistics $U_n$ for $n\to
\infty$, defined in (1.1), with
a general kernel functions $f$ can be reduced to that with degenerate
kernel functions. In the case of unweighted $U$-statistics when
$a(x_1,\dots,x_k)\equiv 1$ the first non-vanishing term in (1.4), i.e.\
the function $f_s$ with the smallest index $s$ in the Hoeffding
representation such that $f_s$ is not identically zero, gives
the dominating contribution to the $U$-statistics. For typical
weighted $U$-statistics the case is similar, but the
general situation is more complex. In this respect we refer to the
second Section of ~[9] and return to this question in Section~2.
 
Our investigation was motivated by a recent paper of
Kevin A.~O'Neil and Richard A.~Redner [9] where asymptotic
distribution of weighted $U$-statistics of degree two was investigated.
This means the investigation of statistics defined by formula (1.1) in
the special case $k=2$. The authors of this paper proved the existence
of a limit distribution with  an appropriate normalization by showing
the convergence of the moments. This method works only if the limit
distribution is determined by its moments. This property holds for
$U$-statistics of degree one or two. But if $k\ge3$, then for
$U$-statistics defined by formula~(1.1) (with a degenerate kernel $f$
satisfying relation ~(1.3)) such a limit distribution appears
which is not determined by its moments. Hence in this case a different
method has to be applied. The aim of the present paper is to find such
a method and to give an explicit expression for the appearing limit.
Let us first explain why  the limit distribution of
$U$-statistics is not determined by its moments for $k\ge3$.
 
 The limit of unweighted (degenerate) $U$-statistics, with
normalization $n^{-k/2}$, can be expressed by means of $k$-fold
Wiener--It\^o integrals with respect to a Wiener process. On the other
hand, the following result is known about the the tail behavior of
multiple stochastic integrals. (See e.g.\  [8] or Section ~6 in~[7].) If
$I_k=\int f(x_1,\dots,x_k)\,B(\,dx_1)\dots\,B(\,dx_k)$ is a $k$-fold
Wiener-It\^o integral with respect to  a Gaussian random measure, then
$$
C_1\exp\left\{-L_1x^{2/k}\right\}<P(|I_k|>x)
<C_2\exp\left\{-L_2x^{2/k}\right\}\quad\text{for all }x>1
$$
with some appropriate constants $C_1>0$, \ $C_2>0$ and $L_1>L_2>0$.
For us the left-hand side of the last inequality is interesting. If a
distribution function $F(x)$ decreases at plus and minus infinity
exponentially fast, then its moments determine its distribution. On the
other hand, if $F(-x)+1-F(x)>C\exp\{-Lx^\alpha\}$ with some
$0\le\alpha<1$ and $C>0$, \ $L>0$ then we cannot say that $F$ is
determined by its moments. (See e.g [2] for an example.) This
second case
appears in the case of $k$-fold stochastic integrals with $k\ge3$.
 
The case of weighted $U$-statistics is similar. The only difference is
that for typical weight functions $a(j_1,\dots,j_k)$ the limit of the
statistics can be  expressed by a $k$-fold stochastic integral with
respect to a Wiener sheet instead of a Wiener process. The Wiener sheet
is the natural two-dimensional analogue of a Wiener process. It is a
two-dimensional Gaussian process $ B(x,y)$, $0\le x,\,y\le 1$, with
expectation zero whose increments
$B(x_2,y_2)+B(x_1,y_1)-B(x_1,y_2)-B(x_2,y_1)$ on disjoint rectangles
$[x_1,x_2]\times[y_1,y_2]$ are independent with variance
$(x_2-x_1)(y_2-y_1)$.
 
Let us briefly explain our approach. First we give a short explanation
about how to handle unweighted $U$-statistics and try to adapt it to
the case of weighted $U$-statistics. Let $F_n(x)$ denote the empirical
distribution function determined by the sample $X_1$,\dots,$X_n$. In the
case of unweighted $U$-statistics when the weight-function $a(\cdot)$
is identically one formula (1.1) can be rewritten as
$$
U_n=\frac{n^k}{k!}\int'f(x_1,\dots,x_k)\,F_n(\,dx_1)\dots
\,F_n(\,dx_k)\;,\tag1.5 $$
where $\int'$ denotes that the hyperplanes $x_i=x_j$ for $i\neq j$ are
cut out from the domain of integration. Since we consider the degenerate
case when relation (1.3) holds, the expression (1.5) does not change if
$F_n(x)$ is replaced by $F_n(x)-x$. We recall that $\sqrt n
\(F_n(x)-x\)\Rightarrow B_0(x)$, where $B_0(x)$ is a Brownian bridge.
Hence, it is natural to expect that we commit a small error by
replacing $\sqrt n \(F_n(x)-x\)$ by $B_0(x)$ or, by
exploiting formula (1.3) again, by a Wiener process $B_0(x)+x\xi$,
where $\xi$ is a standard
normal random variable independent of the Brownian bridge $B_0(x)$. The
last step is useful, because the theory of multiple stochastic integral
is applicable with respect to  Gaussian processes with independent
increments like the Wiener process, but not with respect to a Brownian
bridge. The above argument supplies an informal proof of the limit
theorem for the distribution of unweighted $U$-statistics, and a
rigorous proof can be obtained by justifying the above manipulations.
 
If we want to adapt the above argument to weighted $U$-statistics we
meet some problems at the start. Formula (1.5) does not hold any
longer, moreover $U_n$ cannot be expressed as a functional of $F_n(x)$,
since it is not a function of the ordered sample. But the
above argument can be saved in the special case when the cube
$\{1,\dots,n\}^k$ can be split into finitely many rectangles where the
function $a(j_1,\dots,j_k)$ is equal to a constant. Then limit theorems
for weighted $U$-statistics can be proved in cases when the function
$a(j_1,\dots,j_k)$ can be well approximated by such simple functions. We
shall apply this approach, and throughout the proof we heavily exploit
the $L^2$ isomorphism property of stochastic integrals. We also use
Poissonian approximation, a method which
helped to overcome certain technical difficulties.
The idea that Poissonian approximation is useful for the investigation
of $U$-statistics appeared in the paper of Dynkin and Mandelbaum [1],
and we borrowed it from there.
 
\subheading{2. Formulation of the main results} In this Section we
formulate the main results of this paper. We introduce the following
notation: Given a real number $x$, let $[x]$ denote its integer part.
Our first result is the following
\proclaim{ Theorem 1} \it Let $U_n$ be defined by formula (1.1) with a
function satisfying (1.2) and (1.3). If there
is a continuous function $A(y_1,\dots,y_k)$ on $[0,1]^k$ such that for
$A_n(y_1,\dots,y_k)=a([ny_1],\dots,[ny_k])$ the relation
$$
\lim_{n\to\infty}\int_{[0,1]^k} |A(y_1,\dots,y_k)-A_n(y_1,\dots,y_k)|^2
\,dy_1\dots\,dy_k=0
$$
holds, then the sequence $n^{-k/2}U_n$ tends in distribution to the
stochastic integral
$$
V=\frac1{k!}\int f(x_1,\dots,x_k)A(y_1,\dots,y_k)
\,B(\,dx_1,\,dy_1)\dots B(\,dx_k,\,dy_k)  \;,
$$
where $B(\cdot,\cdot)$ is a Wiener sheet.
\endproclaim
Let us remark that Theorem 1 is not an empty statement. Its condition
can be satisfied for instance if the function $a(j_1,\dots,j_k)$ is
chosen in such a way that its value depends only on the direction of the
vector $(j_1,\dots,j_k)$ in $\Bbb R^k$, and it depends on this direction
continuously. The subsequent Theorems~2 and~3 are natural
generalizations of the results in Section~4 of [9].
\proclaim{ Theorem 2} \it Let $U_n$ be defined by formula (1.1) with
a function satisfying (1.2) and (1.3). Assume that
$a(j_1,\dots,j_k)$ in formula $(1.1)$ can be written in the form
$$
a(j_1,\dots,j_k)=u(h(j_1),\dots,h(j_k))\;,
$$
where $h\colon \bold Z^1\to \{1,\dots,r\}$ with some integer $r$ is
such that the limit
$$
\lim_{n\to\infty}\frac 1n \#\{j,\;j\le n,\,h(j)=s\}=H(s)
$$
exists for all $s=1,\dots,r$, and $u$ is an arbitrary function on
$\{1,\dots,r\}^k$. Then the sequence
$n^{-k/2}U_n$ converges in distribution to the stochastic integral
$$
V=\frac 1{k!}\int f(x_1,\dots,x_k)A(y_1,\dots,y_k) B(\,dx_1,\,dy_1)\dots\,
B(\,dx_k,\,dy_k)\;,
$$
where $B(\cdot,\cdot)$ is a Wiener sheet, and
$$
\align
&A(y_1,\dots,y_k)=u(j_1,\dots,j_k)\\
&\qquad\text{if }
H(1)+\cdots+H(j_s-1)< y_s\le H(j_1)+\cdots+H(j_s),\quad 1\le s\le k\;.
\endalign
$$
\endproclaim
\proclaim  {Theorem 3} \it Let us consider a sequence of random
variables $U_n$ defined by formula (1.1) with a function $f$ satisfying
(1.2) and (1.3) and a weight function of the form
$$
a(j_1,\dots,j_k)=e(j_1)\cdots e(j_k)
$$
with a sequence $e(j)$, $j=1$, 2, \dots, such that the sequence
$e(j)$ is bounded, and the limit
$$
\lim_{n\to\infty}\frac 1n\sum_{j=1}^n e(j)^2=E>0   \tag2.1
$$
exists. Then the random variables $n^{-k/2}U_n$ converge in
distribution to
$$
V=\frac 1{k!}E^{k/2}\int f(x_1,\dots,x_k)W(\,dx_1)\dots W(\,dx_k)\;,
$$
where $W(\cdot)$ is a Wiener process on $[0,1]$.
\endproclaim
Let us remark that, up to a scaling factor, the limit in Theorem~3 is
insensitive to the choice of the sequence $e(j)$.
 
Let us discuss the distribution of the $U$-statistics (1.1) if relation
(1.3) may not hold. We get, by expressing the terms
$f(X_{j_1},\dots,X_{j_k})$ by means of the Hoeffding decomposition
(1.4), that
$$
U_n=\frac1{k!}\sum_{s=0}^k \sum\Sb 1\le j_p\le n,\;1\le p\le s\\
\text{and }j_p\ne j_{p'}\text{ if }p\ne p'\endSb
B_n(j_1,\dots,j_s)f_s(X_{j_1},\dots,X_{j_s}) \tag2.2
$$
with
$$
B_n(j_1,\dots,j_s)=\sum\Sb 1\le l_p\le n,\; 1\le p\le k\\
l_p\ne l_{p'}\text{ if }p\ne p'\\
j_p=l_{r_p} \text{ for some }r_p\quad  p=1,\dots,s\\
\text{such that }1\le r_1<\cdots<r_s\le k \endSb
a(l_1,\dots,l_k)\;, \tag2.3
$$
or by exploiting the symmetry of the function $a(l_1,\dots,l_k)$
$$
B_n(j_1,\dots,j_s)=\binom k s\sum\Sb 1\le l_p\le n,\; 1\le p\le k\\
l_p\ne l_{p'}\text{ if }p\ne p'\\
j_p=l_p \text{ for } p=1,\dots,s\endSb
a(l_1,\dots,l_k)\;. \tag$2.3'$
$$
For unweighted $U$-statistics
$B_n(j_1,\dots,j_s)=\dbinom ks(n-s-1)\cdots (n-k)\asymp\dbinom ks
n^{k-s}$.
The orthogonality of the random variables $f_s(X_{j_1},\dots,X_{j_s})$
together with this relation imply that the the inner sum with the
smallest index $s$ for which $f_s$ does not vanish identically gives the
dominating contribution to the external sum in (2.2), and it has order
$n^{k-s/2}$. For typical weighted $U$-statistics a similar picture
arises. But since the coefficients $a(j_1,\dots,j_k)$ may cause some
additional cancellation, the situation is more complex. We show this in
an example which may be of special interest. We consider the model in
Theorem~3, but do not assume that the kernel function $f$ defines
degenerate statistics. We consider statistics of the form
$$
U_n=\sum_{1\le j_1<j_2\dots<j_k\le n}
e(j_1)\cdots e(j_k)f(X_{j_1},\dots,X_{j_k})\;. \tag 2.4
$$
Put
$$
F_n=n^{-1/2}\sum_{j=1}^n e(j)\;.\tag2.5
$$
The limit behavior of $U_n$ is different in the cases when $F_n$
has a finite non-zero limit and when it tends to zero or to infinity.
We describe the case when $F_n$ has a finite non-zero limit. This seems
to be the most interesting case, when the contributions of different
terms in the Hoeffding representation have the same order and the
limit can be represented as a sum of stochastic integrals of different
multiplicity. This question was considered in a special case in papers
[4] and [9],  and it also shows some analogy with the surface charge
in~[6]. The remaining cases will be only briefly discussed.
\proclaim {Theorem 4} \it Let us consider the weighted $U$-statistics
defined in (2.4) with a bounded sequence $e(j)$, $j=1$, 2, \dots,
satisfying (2.1) and a square integrable kernel function $f$. Assume
that the sequence $F_n$ defined in (2.5) has a limit
$\lim\limits_{n\to\infty}F_n=F$.  Let us
take the Hoeffding decomposition of the function $f$ given in formulas
(1.4) and $(\text{1.4}')$. Then the sequence $n^{-k/2}U_n$ converges in
distribution to the sum of stochastic integrals
$$
\frac{D_kf_0}{k!}+ \sum_{s=1}^k\frac{D_{k-s}}{s!(k-s)!}E^{s/2}\int
f_s(x_1,\dots,x_s)\,W(\,dx_1)\dots\,W(\,dx_s)
$$
as $n\to\infty$, where $W(x)$ is a Wiener process in the interval
$[0,1]$, and the sequence $D_s$ is defined by the following recursive
formula: $D_0=1$, $D_1=F$, and
$$
D_s=F^s-\sum_{p=1}^{\[\frac s2\]} \frac{s!}{2^pp!(s-2p)!}E^p D_{s-2p}\;.
$$
\endproclaim
\subheading{3. Approximation of $U$-statistics} In this   section we
approximate weighted $U$-statistics with polynomials of independent
centered Poissonian random variables (by a centered Poissonian random
variable we mean a Poissonian random variable minus its expectation) and
show that  a small error is committed if these centered Poissonian
random variables are replaced by independent Gaussian random variables.
To formulate these results we introduce some definitions and remarks.
\demo{Remark 1} For a function $f$ satisfying (1.1) and any $\e>0$ an
approximating step-function
$g(x_1,\dots,x_k)=g_\e(x_1,\dots,x_k)$ can be given such that
$$
\int_{[0,1]^k} |f(x_1,\dots,x_k)- g(x_1,\dots,x_k)|^2
\,dx_1\dots\,dx_k<\e\;, \tag3.1
$$
and there is some integer $L=L(\e)$ such that the function
$g(x_1,\dots,x_k)$
is constant on all cubes $\(\dfrac {j_1-1}L,\dfrac {j_1}L\]\times\cdots
\times\(\dfrac {j_k-1}L,\dfrac {j_k}L\]$, $1\le j_s\le L$ for
$s=1,\dots,k$, and it is zero on those cubes for which $j_s=j_{s'}$ with
some $s\ne s'$.
\enddemo
We introduce the notion of $\e$-approximability of a weight function
$a(j_1,\dots,j_k)$.
\proclaim {Definition of $\e$-approximation of weight functions} \it
A sequence
$a(j_1,\dots,j_k)$ is $\e$-approximable by a set of elementary functions
$b_n^\e(j_1,\dots,j_k)$, $1\le j_s\le n$, $1\le s\le k$ if
$$
n^{-k}\sum\left|a(j_1,\dots,j_k)-b_n^\e(j_1,\dots,j_k)\right|^2<\const \e
$$
with some constant independent of $n$ and $\e$, and the function
$b_n^\e(j_1,\dots,j_k)$ has the following property:
 
There exists a partition $\Lambda_1=\Lambda_1(n,\e),\dots,
\Lambda_p=\Lambda_p(n,\e)$ of the set $\{1,\dots,n\}$ with cardinality
$|\Lambda_1|=N_1=N_1(n,\e)$,\dots, $|\Lambda_p|=N_p=N_p(n,\e)$ with some
number $p=p(\e)$ which may depend on $\e$ but not on $n$ and numbers
$B_n^\e(m_1,\dots,m_k)$ whose absolute values are bounded by some number
$B(\e)$ which does not depend on~$n$, $1\le m_s\le p$, $s=1,\dots,k$,
such that   %@szovegvaltoztatas
$$
b_n^\e(j_1,\dots,j_k)= B_n^\e(m_1,\dots,m_k)\quad\text{if }
j_s\in\Lambda_{m_s}\text{ for all }\;1\le s\le k\;.\tag3.2
$$
We shall say that the above $\e$-approximation is determined at level
$n$ by the partition $\Lambda_1,\dots,\Lambda_p$ of the set
$\{1,\dots,n\}$ and the function $B_n^\e(m_1,\dots,m_k)$.
\endproclaim
Now we formulate the  results of this Section.
\proclaim{Lemma 1}\it Let $U_n$ be a weighted $U$-statistic as defined
in
(1.1) with a kernel function $f$ satisfying (1.2) and (1.3) and a weight
function $a(j_1,\dots,j_k)$ which is $\e$-approximable. Let this
$\e$-approximation be determined at level $n$ by a partition
$\Lambda_1,\dots,\Lambda_p$ of the set $\{1,\dots,n\}$ and a function
$B_n^\e(m_1\dots,m_k)$. Take an $\e$-approximating step function
$g(x_1,\dots,x_k)=g_\e(x_1,\dots,x_k)$ of the function $f$ which
satisfies the properties formulated in Remark~1  and put
$$
g^*(l_1,\dots,l_k)=g\(\frac {l_1}L,\dots,\frac{l_k}L\)\;,\tag3.3
$$
where $L$ is the same as in Remark~1. A set of
independent centered Poissonian random variables $\eta_{m,l}$, $1\le
m\le p$ and $1\le l \le L$, can be constructed  with parameter $\dfrac
{N_m}L$ ($N_m$ is the cardinality of the set $\Lambda_m$) such that
$$
\align
&E\biggl|n^{-k/2}\biggl(U_n-\frac1{k!}\sum\Sb m_s=1,\dots,p\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\eta_{m_1,l_1}
\dots\eta_{m_k,l_k}\biggr)\biggr|^2\\
&\qquad<\const\(\e+\frac{C(\e,k)}{\sqrt n}\)
\endalign
$$
with some constant $C(\e,k)$ depending only on $\e$ and $k$.
\endproclaim
\proclaim{Lemma 2} \it Let us fix some positive integers $p$ and $k$.
Let us have for all positive integers $n$  a sequence of independent
centered Poissonian random variables $\eta_s=\eta_s(n)$, with parameter
$N_s$ and a sequence of independent Gaussian random variables
$\xi_s=\xi_s(n)$ with expectation zero and variance $N_s$, $1\le
s\le p$, such that $N_s\le n$, $1\le s\le p$. Consider the polynomials
$$
\align
S_n&=n^{-k/2}\sum\Sb j_s=1,\dots,p\\ j_s\ne j_{s'} \text{ if }s\ne
s'\endSb b_n(j_1,\dots,j_k)\,\eta_{j_1}\cdots\eta_{j_k}\\
\intertext{and}
T_n&=n^{-k/2}\sum\Sb j_s=1,\dots,p\\ j_s\ne j_{s'} \text{ if }s\ne
s'\endSb b_n(j_1,\dots,j_k)\,\xi_{j_1}\cdots\xi_{j_k}
\endalign
$$
with coefficients satisfying the relation
$$
|b_n(j_1,\dots,j_k)|<K\quad\text{for all }1\le j_s\le p,\quad1\le s\le k
$$
with some positive constant $K$. Then for all $t\in \Bbb R^1$
$$
\lim_{n\to\infty}\(Ee^{itS_n}-Ee^{itT_n}\)=0\;.
$$
\endproclaim
\demo{Proof of Lemma 1} Introduce the expression
$$
U_n^{(1)}=\sum_{1\le j_1<j_2\dots<j_k\le n}
b_n^\e(j_1,\dots,j_k)f(X_{j_1},\dots,X_{j_k})\;,
$$
where $b_n^\e$ is the $\e$-approximating sequence of $a_n$. Since
$$
Ef(X_{j_1},\dots,X_{j_k})f(X_{j_1'},\dots,X_{j_k'})=0 \quad\text{if }
(j_1,\dots,j_k)\ne (j_1',\dots,j_k')
$$
by (1.3), hence
$$
En^{-k}(U_n-U^{(1)}_n)^2=
n^{-k}\sum\left|a(j_1,\dots,j_k)-b_n^\e(j_1,\dots,j_k)\right|^2<\const
\e\;. \tag3.4
$$
Let $\nu_1,\dots,\nu_p$ be independent Poisson distributed random
variables with parameter $N_s$, $1\le s\le p$, independent of the random
variables $X_j$, $j=1,\dots, n$, too. Let the sets $\Lambda_m$ appearing
in the definition of $\e$-approximability be
$\Lambda_m=\{s_1(m),\dots,s_{N_m}(m)\}$,
$s_1(m)<\cdots<s_{N_m}(m)$, $m=1,\dots,p$.
We define sets $\Lambda'_m$,
with (random) size $\nu_m$, $1\le m\le p$, which are close to the sets
$\Lambda_m$. Put $\Lambda'_m=\{\bar s_1(m),\dots,\bar s_{\nu_m}(m)\}$,
$\bar s_1(m)<\cdots<\bar s_{\nu_m}(m)$ such that $\bar s_l(m)=s_l(m)$
for $l\le \min (N_m,\nu_m)$ and $\bar s_l(m)=J(m)+l-N_m$ with
$J(m)=\sum\limits_{p=1}^{m-1}(\nu_p-N_p)_+$ if $N_m<l\le \nu_m$. We
consider a set of independent random variables $Y_{m,l}$, $1\le m\le p$
and $1\le l\le \nu_m$, with uniform distribution in the interval $[0,1]$
which are independent of the random variables $\nu_m$, $n=1,\dots,p$,
and also have the property: $$
Y_{m,l}=X_{s_l(m)}\quad\text{if }l\le\min (\nu_m, N_m),\quad m=1,\dots,
p\;.
$$
(The choice of the random variables $Y_{m,l}$ with such properties is
possible. They must be chosen conditionally independent and uniformly
distributed on $[0,1]$ under the condition that the values of the random
variables $\nu_m$ are prescribed.) Define the numbers $l(j)$ and $m(j)$
as the indices such that $j\in\Lambda'_{m(j)}$ and $j=\bar s_{l(j)}(m(j))$
if $j$ is an element of some $\Lambda'_m$, $1\le m\le p$. Otherwise let
$l(j)$ and $m(j)$ be meaningless.
 For $1\le m_i\le p$ and $l_i=1$, 2, \dots, $i=1,\dots,k$ put
$$
\bar b_n^\e\((m_1,l_1),\dots,(m_k,l_k)\)=\left\{
\aligned
& B_n^\e(m_1,\dots,m_k)\quad\text{if }l_i\le \nu_{m_i},
\quad i=1,\dots,k \\
&0\quad\text{otherwise}
\endaligned \right.
$$
and
$$
\align
U_n^{(2)}=
\sum_{1\le j_1<j_2\dots<j_k<\infty}
&\bar b_n^{(\e)}\((m(j_1),l(j_1)),\dots,((m(j_k),l(j_k))\)\\
&\qquad \times f\(Y_{m(j_1),l(j_1)},\dots,Y_{m(j_k),l(j_k)}\)\;.
\endalign
$$
We define
$\bar b_n^{(\e)}((m(j_1),l(j_1))\dots,((m(j_k),l(j_k)))=0$ in the last
expression if $m(j_s)$ and $l(j_s)$ are not defined for some ~$s$. We
claim that
$$
E\(n^{-k/2}(U_n^{(1)}-U_n^{(2)})\)^2< {C(\e,k)}n^{-1/2}\;.
\tag3.5
$$
 
To prove relation (3.5) observe that the number of terms which appear
in the sum
$U_n^{(1)}$ but not in $U_n^{(2)}$ (two terms in these sums
agree if the function
$f$ is a function of the same random variables in them, and it has the
same coefficient) and the number of terms which appear in $U_n^{(2)}$,
but not in $U_n^{(1)}$ is less than
$$
k\max_{s\le p}\left|\nu_s-N_s\right|\(\max_{s\le p} N_s^{k-1}+\max_{s\le
p} \nu_s^{k-1}\)\;.
$$
This relation together with the orthogonality relations imply that
$$
E\(n^{-k/2}(U_n^{(1)}-U_n^{(2)})\)^2
<\frac{C(\e,k)}{n^k}
E\(\max_{s\le p}\left|\nu_s-N_s\right|(\max_{s\le p}
N_s^{k-1}+\max_{s\le p} \nu_s^{k-1})\)\;.\tag3.6
$$
Since
$$
E\nu_s^L\le C(L)N_s^L\le C(L)n^L
$$
and
$$
E|\nu_s-N_s|^2\le n
$$
with some $C(L)>0$ for any $s\le p$ and $L\ge1$, hence by the Schwartz
inequality
$$
\align
&\[E\(\max_{s\le p}\left|\nu_s-N_s\right|(\max_{s\le p}
N_s^{k-1}+\max_{s\le p} \nu_s^{k-1}\)\]^2\\
&\qquad\;\le E\max |\nu_s-N_s|^2\cdot E\[\max
N_s^{k-1}+\max\nu_s^{k-1}\]^2\le \const n^{2k-1}.
\endalign
$$
The last inequality together with (3.6) imply (3.5).
 
Put $\Sigma=\Sigma(\e)=[0,1]\times\{1,\dots,p\}$, and define the random
field consisting of the points $Z(m,l)=(Y_{m,l},m)$, $1\le m\le p$ and
$1\le l\le\nu_m$ on it. ($\Sigma$
depends on $\e$ through $p=p(\e)$.) Then $Z(m,l)$ is a Poisson process
such that the expected value of the points $Z(\cdot,\cdot)$ in a set
$\bigcap\limits_{m=1}^p (A_m,m)\subset\Sigma$ equals
$\sum\limits_{m=1}^p
N_m\lambda(A_m)$, where $\lambda(\cdot)$ denotes the Lebesgue
measure. Introduce the counting measure $\mu_n=\mu_n^\e$ on $\Sigma$
such that $\mu_n(B)$  is the number of points $Z(\cdot,\cdot)$ in the
set $B$ for $B\subset\Sigma$. Let $P_n$ be its centering, i.e.\
$P_n(B)=\mu_n(B)-E\mu_n(B)$. Given a function $f(x_1,\dots,x_k)$ on
$[0,1]^k$ define the function $f^\e_{\bar b}((x_1,m_1),\dots,(x_k,m_k))$
on $\Sigma^k$ as
$$
f^\e_{\bar
b}((x_1,m_1),\dots,(x_k,m_k))=B_n^\e(m_1,\dots,m_k)f(x_1,\dots,x_k)\;,
$$
where the function $B_n^\e$ is the same as that which appears in the
definition
of $\e$-approximability of a weight function. Then $U_n^{(2)}$ can be
rewritten as
$$
U_n^{(2)}=\frac1{k!}\int'_{\Sigma^k} f^\e_{\bar
b}(z_1,\dots,z_k)\,\mu_n(dz_1)\dots\,\mu_n(dz_k)
$$
with $z_s=(x_s,m_s)$, $x_s\in [0,1]$ and $m_s\in\{1,\dots,p\}$ for
$s=1,\dots,k$, where $\int'$ means that the hyperplanes $z_j= z_{j'}$
for $j\ne j'$ are cut out from the domain of integration. Condition
($1.4'$) also implies that %@vesszo
$$
\int_\Sigma f_{\bar b}^\e(z,z_2,\dots,z_k)\,\bar\lambda(dz)=0\quad\text
{for all }z_2,\dots,z_k
$$
with $\bar\lambda(A)=E\mu_n(A)$ for $A\subset \Sigma$. Hence
$$
U_n^{(2)}=\frac1{k!}\int'_{\Sigma^k} f^\e_{\bar
b}(z_1,\dots,z_k)\,P_n(dz_1)\dots\,P_n(dz_k)\;.\tag3.7
$$
 
Define the mapping $I$ from the set of function $f^\e_{\bar b}$ to the
space of random variables on $(\Omega,\Cal A,P)$, where
$(\Omega,\Cal A,P)$ is the probability space where the Poisson process
is defined, as
$$
I(f^\e_{\bar b})=\frac 1{\sqrt{k!}}\int'_{\Sigma^k}
f^\e_{\bar b}(z_1,\dots,z_k)\,P_n(dz_1)\dots P_n(dz_k)\;.
$$
It is known in the theory of Poissonian integrals, and actually it is
not difficult to prove that
$$
\int f^\e_{\bar b}(z_1,\dots,z_k)^2\;\bar\lambda(dz_1)\dots\,\bar
\lambda(dz_k)=EI(f^\e_{\bar b})^2\;.
$$
Let $g(x_1,\dots,x_k)=g_\e(x_1,\dots,x_k)$ be an approximating
function of $f$ having the properties mentioned in
Remark~1. Since $N_s\le n$ for all $1\le s\le p$,
$\bar\lambda(A)\le n\lambda (A)$ for $A\subset \Sigma$, where
$\lambda(\cdot)$ denotes the Lebesgue measure on $\Sigma$. This fact
together with (3.1) and the definition of $g^\e_{\bar b}$ imply that
$$
\int\left |f^\e_{\bar b}(z_1,\dots,z_k)-g^\e_{\bar
b}(z_1,\dots,z_k)\right|^2\bar\lambda(\,dz_1)\dots\bar\lambda(\,dz_k)
<\const \e n^k\;.
$$
The last relation together with (3.7) and the $L^2$ isomorpism of the
mapping ~$I$ (applying it for $f-g$) imply that
$$
n^{-k}E\[U_n^{(2)}-\frac1{k!}\int'_{\Sigma^k} g^\e_{\bar
b}(z_1,\dots,z_k)\,P_n(dz_1)\dots\,P_n(dz_k)\]^2\le\const\e\;.
$$
This relation together with (3.4) and (3.5) give that
$$
n^{-k}E\[U_n-\frac1{k!}\int'_{\Sigma^k} g^\e_{\bar
b}(z_1,\dots,z_k)\,P_n(dz_1)\dots\,P_n(dz_k)\]^2\le
\(\const\e+\frac{C(\e,k)}{\sqrt n}\)\;.\tag3.8
$$
 
The random measure $P_n\(\(\dfrac{l-1}L,\dfrac lL\], m\)$ is a centered
Poissonian random variable with parameter $\dfrac{N_m}L$, and the
measures
of the sets $\(\(\dfrac{l-1}L,\dfrac lL\], m\)$ are independent for
different pairs $(l,m)$. Hence
$$
\align
&\int'_{\Sigma^k}g^\e_{\bar b}(z_1,\dots,z_k)
\,P_n(dz_1)\dots\,P_n(dz_k)\\
&\qquad=\sum\Sb m_s=1,\dots,p\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\eta_{m_1,l_1}
\cdots\eta_{m_k,l_k}\;,
\endalign
$$
and relation (3.8) implies Lemma~1. \qed \enddemo
\demo{Proof of Lemma 2} Since
$$
\left|\exp\(i\sum a_j\)-\exp\(i\sum b_j\)\right|\le
\sum|\exp(ia_j)-\exp(ib_j)|\;,
$$
hence
$$
\align
&\left|E\exp\{itS_n\}-E\exp\{itT_n\}\right|\\
&\qquad\le\const\sup\Sb {|s|\le
K|t|}\\j_1\dots,j_k\endSb
\left|E\exp\left\{is\frac{\eta_{j_1}}{\sqrt
n}\cdots\frac{\eta_{j_k}}{\sqrt n}\right\}-
E\exp\left\{is\frac{\xi_{j_1}}{\sqrt n}\cdots\frac{\xi_{j_k}}{\sqrt
n}\right\}\right| \tag3.9
\endalign
$$
with some $K>0$. We may assume that
$$
\sup_{j\le p}E\left|n^{-1/2}(\eta_j(n)-\xi_j(n))\right|^2\to
0\quad \text{as }n\to\infty\;.\tag3.10
$$
Indeed, if $\dfrac{\eta_j}{\sqrt{N_j}}$ is the quantile transform of
$\dfrac{\xi_j}{\sqrt{N_j}}$, i.e.\
$$
\dfrac{\eta_j}{\sqrt{N_j}}=F_j^{-1}\(\Phi\(\dfrac{\xi_j}{\sqrt{N_j}}\)\)\;,
$$
where $\Phi$ is the standard
normal distribution function, $F_j$ is the distribution function of
$\dfrac{\eta_j}{\sqrt {N_j}}$ and $N_j$ is the variance of $\xi_j$ and
~$\eta_j$, then it is not difficult to see with the
help of the central limit theorem that (3.10) holds for this $\xi_j$ and
$\eta_j$. (Actually the
following stonger estimate holds. See formula (2.6) in Lemma~1 of~[5].)
$$
E\left|
n^{-1/2}(\eta-\xi)\right|^2\le \const \frac 1n\;. $$
On the other hand, the random variable $S_n$  defined with these random
variables $\eta_j$ has the right distribution. Then we have
 
$$
\align
&\left|E\exp\left\{is\frac{\eta_{j_1}}{\sqrt
n}\cdots\frac{\eta_{j_k}}{\sqrt n}\right\}-
E\exp\left\{is\frac{\xi_{j_1}}{\sqrt n}\cdots\frac{\xi_{j_k}}{\sqrt
n}\right\}\right|\\
&\qquad\le n^{-k/2} |s|\, E\left|{\eta_{j_1}}
\cdots{\eta_{j_k}}-
{\xi_{j_1}}\cdots{\xi_{j_k}}\right| \\
&\qquad\le
n^{-k/2}|s|\,\sum_{p=0}^{k-1}E\left|\eta_{j_1}\cdots\eta_{j_p}\right|\,
\left|\eta_{j_{p+1}}-\xi_{j_{p+1}}\right|\,\left|\xi_{j_{p+2}}
\cdots\xi_{j_k}\right| \\
&\qquad= n^{-k/2}|s|\,\sum_{p=0}^{k-1}E\left|\eta_{j_1}\right|\cdots
E\left|\eta_{j_p}\right|\,
E\left|\eta_{j_{p+1}}-\xi_{j_{p+1}}\right|\,E\left|\xi_{j_{p+2}}\right|
\cdots E\left|\xi_{j_k}\right| \\ &\qquad\le
n^{-1/2}|s|\,\const\sum_{p=0}^{k-1}E\left|\eta_{j_{p+1}}
-\xi_{j_{p+1}}\right| \\
&\qquad\le\const \sup E\left|n^{-1/2}(\eta_j(n)-\xi_j(n))\right|^2
\endalign
$$
because of the independence of the pairs $(\eta_j(n),\xi_j(n))$ and the
condition $N_j\le n$. The last relation together with (3.10) imply that
the right-hand side of (3.9) tends to zero, hence Lemma~2 holds.
\qed \enddemo
\subheading{4. Proof of the Theorems}
\demo{Proof of Theorem 1} There is a step function $A^\e(y_1,\dots,y_k)$
such that
$$
\int_{[0,1]^k}\left
|A_n(y_1,\dots,y_k)-A^\e(y_1,\dots,y_k)\right|^2\,dy_1\dots,\,dy_k
$$
for $n>n(\e)$, and it has the following structure: There is some $T>0$
such that
$$
\align
A(y_1,\dots,y_k)&=A^\e\(\frac{m_1}T,\dots,\frac {m_k}T\)\quad\text {if }
\frac{m_s-1}T<y_s\le \frac{m_s}T\\
&\qquad\qquad\text{ and all numbers $m_1$,\dots, $m_k$ are different}\\
&=0\quad \text{if there is some }1\le s<s'\le k\text{ and }0<m\le T \\
&\qquad\qquad \text{such that }\frac{m-1}T< y_s, y'_s\le\frac m T\;.
\endalign
$$
There is an $\e$-approximation of the function $a(m_1,\dots,m_k)$ which
is determined at  level $n>n(\e)$ by the partition $\Lambda_m=\(\[\dfrac
{m-1}T n\],\[\dfrac m Tn\]\]$, $1\le m\le T$, and the functions
$$
B^\e_n(m_1,\dots,m_k) =A^\e\(\frac{m_1}T,\dots,\frac {m_k}T\)\;.
$$
Let $g(x_1,\dots,x_k)=g_\e(x_1,\dots,x_k)$ be an $\e$-approximating step
function of $f$ which satisfies Remark~1. Let the function
$g^*(l_1,\dots,l_k)$  be defined by (3.3) and the above function $g$. We
get by Lemma~1 that for
$$
S_n=\frac1{k!}n^{-k/2}\sum\Sb m_s=1,\dots,T\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\eta_{m_1,l_1}
\dots\eta_{m_k,l_k}
$$
$$
E(n^{-k/2}U_n-S_n)^2\le\const\(\e+\frac{C(\e,k)}{\sqrt n}\)\;,\tag4.1
%@^{-k/2}beillesztese
$$
where $\eta_{m,l}$, $1\le m\le T$ and $1\le l\le L$, are
appropriate independent
centered Poissonian random variables with parameter $\dfrac nT$.
 
On the other hand,
$$
\align
\int_{[0,1]^{2k}}&\left|f(x_1,\dots,x_k)A(y_1,\dots,y_k)
-g_\e(x_1,\dots,x_k)A^\e(y_1,\dots,y_k)\right|^2\, \\
&\qquad\qquad  dx_1\,dy_1\dots\,dx_k \,dy_k \le \const\e \;,
\endalign
$$
and because of the $L^2$ isomorphism of Wiener-It\^o integrals
$$
E\(V-T_n\)^2\le\const \e\;,\tag4.2
$$
where $V$ is the stochastic integral  with the limit distribution
defined in the formulation of Theorem~1, and
$$
T_n=\frac1{k!}n^{-k/2}\sum\Sb m_s=1,\dots,T\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\xi_{m_1,l_1} \dots\xi_{m_k,l_k}
$$
with independent Gaussian random variable $\xi_{m,l}$, $1\le m\le T$
and $1\le l\le L$, with expectation zero and variance $\dfrac nT$.
 
It follows from (4.1) that
$$
\align
&\left|E\exp\{itn^{-k/2}U_n\}-E\exp\{itS_n\}\right|\le |t|
E|n^{-k/2}U_n-S_n| \\
&\qquad\le |t|\(E(n^{-k/2}U_n-S_n)^2\)^{1/2}
\le\const\(\e^{1/2}+C(\e,k)n^{-1/4}\)
\endalign
$$
for any $t\in\Bbb R^1$. Similarly, it follows from (4.2) that
$$
\left|Ee^{itV}-Ee^{it T_n}\right|\le \const\e^{1/2}\;.
$$
Since $Ee^{itS_n}-Ee^{itT_n}\to 0$ by Lemma~2 the last two relations
imply that
$$
\limsup_{n\to\infty}\left| E\exp\{itn^{-k/2}U_n\}-E\exp\{it
V\}\right|\le \const\e^{1/2}
$$
Since the last relation holds for any $\e>0$ we get that
the characteristic function of $U_n$ satisfies the relation
$$
E\exp\{itn^{-k/2}U_n\}\to E\exp\{it V\}\quad\text {for all }t\in\Bbb
R^1\;. $$
The last relation implies Theorem~1.\qed
\enddemo
\demo{Proof of Theorem 2} The proof is similar to that of Theorem~1.
Now we can choose the function $a(j_1,\dots,j_k)$ itself as its
approximation by elementary function. Then this approximation is
determined at level $n$ by the sets
$$
\Lambda_m=\{j;\quad1\le j\le n,\;h(j)=m\}\,\quad m=1,\dots,r\;,
$$
and the function $B^\e_n(m_1,\dots,m_k)=u(m_1,\dots,m_k)$. Then
$N_m=N_m(n)$, the cardinality of the set $\Lambda_m$, satisfies the
relation
$$
\lim_{n\to\infty}\frac {N_m(n)}n=H(m)\quad\text{for
}m=1,\dots,r\;.\tag4.3
$$
Let $g=g_\e$ be an approximating step function of $f$ satisfying
Remark~1, and let the function $g^*$ be defined by (3.3). Then
$$
S_n=\frac1{k!}n^{-k/2}\sum\Sb m_s=1,\dots,r\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\cdots,l_k)\eta_{m_1,l_1}
\dots\eta_{m_k,l_k}
$$
well approximates $n^{-k/2}U_n$ in $L^2$ norm, where $\eta_{m,l}$ are
independent
centered Poissonian random variables with parameter $\dfrac{N_m}L$.
Because of the definition of the function $A(y_1,\dots,y_k)$ and (4.3)
the stochastic integral $V$ appearing in Lemma~2 can be well
approximated in $L^2$ norm by
$$
T_n=\frac1{k!}n^{-k/2}\sum\Sb m_s=1,\dots,r\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\xi_{m_1,l_1}
\cdots\xi_{m_k,l_k}\;,
$$
where $\xi_{m,l}$ are independent Gaussian random variables with
expectation zero and variance $\dfrac{N_m}L$. Then Lemma 2 implies that
the characteristic functions of $S_n$ and $T_n$ are close to each other.
Then a natural adaptation of the argument in the proof of Theorem~1
implies that the characteristic function of $n^{-k/2}U_n$ tends to that
of $V$, and this implies Theorem~2. \qed
\enddemo
In the proof of Theorem~3 we need a lemma which shows why the sequence
$e(j)$  influences only the norming constant of the limit
distribution of $U_n$ in Theorem~3.
\proclaim{Lemma 3} \it Let $f(x_1,\dots,x_k)$ be a square integrable
function on $[0,1]^k$, $h(y)$ a function on $[0,1]$ such that
$\int_0^1 h^2(y)\,dy=1$, $W(x)$ a Wiener process on $[0,1]$ and $B(x,y)$
a Wiener sheet on $[0,1]^2$. Then the stochastic integrals
$$
\align
I_1&=\int f(x_1,\dots,x_k)\,W(\,dx_1)\dots\,W(\,dx_k)\\
\intertext{and}
I_2&=\int f(x_1,\dots,x_k)h(y_1)\cdots h(y_k)
\,B(\,dx_1,\,dy_1)\dots\,B(\,dx_k,\,dy_k)
\endalign
$$
have the same distribution.
\endproclaim
\demo{Proof of Lemma 3} This lemma could have been proved by considering
first elementary functions and then approximating general functions by
them. We choose a different way. We express both $I_1$ and $I_2$ by
means of It\^o's formula as a series of independent Gaussian random
variables and observe that these two expressions have the same
distribution.
 
Let $\psi_1$, $\psi_2$,\dots be a complete orthonormal system in
$[0,1]$, and take the expansion
$$
f(x_1,\dots,x_k)=\sum c(j_1,\dots,j_k)\psi_{j_1}(x_{1})\cdots
\psi_{j_k}(x_{k})\;.
$$
The functions $\varphi_j(x,y)=\psi_j(x)h(y)$, $j=1$, 2,\dots, are
orthonormal in $[0,1]^2$,
and
$$
f(x_1,\dots,x_k)h(y_1)\cdots h(y_k)=\sum
c(j_1,\dots,j_k)\varphi_{j_1}(x_{1},y_1)\cdots
\varphi_{j_k}(x_{k,},y_k)\;. $$
By It\^o's formula (see [3], or~[7], Section~7) these relations imply
that
$$
\align
I_1&=\sum c(j_1,\dots,j_k):\!\eta_{j_1}\cdots \eta_{j_k}\!: \tag4.4\\
\intertext{and}
I_2&=\sum c(j_1,\dots,j_k):\!\zeta_{j_1}\cdots \zeta_{j_k}\!: \tag$4.4'$
\endalign
$$
with $\eta_j=\int \psi(x)\,W(\,dx)$ and $\zeta_j=\int
\varphi(x,y)\,B\,(dx,\,dy)$. Here $:\!\eta_{j_1}\cdots
\eta_{j_k}\!\!:$, the Wick polynomial of the corresponding product,
equals $\prod H_{l_m}(\eta_m)$, where $l_m$ denotes the multiplicity of
the index~$m$ in the set $\{j_1,\dots,j_k\}$ and $H_m(x)$ is the $m$-th
Hermite polynomial. The definition of $:\!\zeta_{j_1}\cdots
\zeta_{j_k}\!\!:$ is similar. Since both sequences $\eta_j$ and
$\zeta_j$, $j=1$, 2,\dots, are sequences of independent standard normal
random variables, the expressions in (4.4) and ($4.4'$) have the same
distributions. Lemma~3 is proved. \qed \enddemo
\demo{Proof of Theorem 3} The proof is similar to that of Theorems~1
and~2. Let us fix some small $\e>0$, and define the sequence $\bar
e(j)=\bar e^\e(j)$, $j=1$, 2,\dots, by the formula
$$
\bar e(j)=K\e,\quad \text{if } K\e\le e(j)<(K+1)\e\quad \text{with some
integer }K\;.
$$
Then
$$
\left|\frac 1n\sum_{j=1}^n \bar e^2(j)-\frac 1n\sum_{j=1}^n e^2(j)
\right|\le \const\e\;,     \tag4.5
$$
and the sequence $\bar e(j)$, $j=1$, 2,\dots takes finitely many values
$K_1\e<K_2\e<\cdots<K_p\e$ with some $p=p(\e)$ because of the
boundedness of the sequence $e(j)$. Let the sequence $\bar e(j)$,
$j=1,\dots,n$, take the value $K_l\e$ $N_l=N_l(n)$ times, $1\le l\le p$.
Introduce the function $h_n(y)=h_n^\e(y)$ on $[0,1]$ as
$$
h_n(y)= K_p\e\quad\text{on the interval }
\frac 1n\sum_{l=1}^{p-1}N_l<y\le \frac 1n\sum_{l=1}^{p}N_l
$$
and the number $E(n)=E^\e(n)=\int_0^1 h_n^2(y)\,dy$. By Lemma~3 the
stochastic integral
$$
V_n=\frac1{k!} \(\frac E{E(n)}\)^{k/2}\!\int
f(x_1,\dots,x_k)h_n(y_1)\cdots
h_n(y_k) B(\,dx_1,dy_1)\dots B(\,dx_k,dy_k) \tag4.6
$$
has the same distribution as the stochastic integral $V$ defined in
the formulation of Theorem~3.
 
The sequence $a(j_1,\dots,j_k)=e(j_1)\cdots e(j_k)$ can be
$\e$-approximated by elementary functions such that this approximation
is determined at level $n$ by the partition
$$
\Lambda_m=\{j;\;\;1\le j\le n,\; \bar e(j)=K_{m}\e\}\quad\text{for
}1\le m\le p
$$
and the function $B^\e_n(m_1,\dots,m_k)=K_{m_1}\e\cdots K_{m_k}\e$.
 
Let $g_\e(x_1,\dots,x_k)=g(x_1,\dots,x_k)$ be an approximating step
function of $f$ satisfying Remark~1. Then the random variables
$n^{-k/2}U_n$ can be well approximated in $L^2$ norm by
$$
S_n=\frac1{k!}n^{-k/2}\sum\Sb m_s=1,\dots,p\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\eta_{m_1,l_1}
\dots\eta_{m_k,l_k}
$$
by Lemma 1, where $\eta_{m,l}$ are independent
centered Poissonian random variables with parameter $\dfrac{N_m}L$.
Because of the $L^2$ isomorphism property of Wiener-It\^o integrals the
random variable $V_n$ defined in (4.6) is well approximated in $L^2$
norm by
$$
T_n=\frac1{k!}\(\frac E{E(n)}\)^{k/2}\!n^{-k/2}\!\sum\Sb m_s=1,\dots,p\\
l_s=1,\dots,L \\ \text{for }s=1,\dots,k\endSb
\! B_n^\e(m_1,\dots,m_k) g^*(l_1,\dots,l_k)\xi_{m_1,l_1}
\dots\xi_{m_k,l_k}\;,
$$
where $\xi_{m,l}$ are independent Gaussian random variables with
expectation zero and variance $\dfrac{N_m}L$. Since
$$
\lim_{\e\to 0}\sup_n |E^\e(n)-E|=0
$$
by (4.5), Lemma 2 implies that the characteristic functions of $S_n$ and
$T_n$ are close to each other. These relations together with the
observation that $V_n$ and $V$ have the same distribution, and this
implies the proof of Theorem~3 similarly to the proof of Theorem~1.\qed
\enddemo
 
The proof of Theorem 4 is based on the following multidimensional
version of Theorem~3 and a lemma about the asymptotic behavior of the
expression $B_n(j_1,\dots,j_k)$ defined in (2.3).
\proclaim{Theorem $\bold 3'$}\it Consider the random variables $$
U_n^{(s)}=\sum_{1\le<j_1<j_2\dots<j_s\le n}
e(j_1)\cdots e(j_s)f_s(X_{j_1},\dots,X_{j_s}),\quad 1\le s\le k\;,
$$
with a sequence $e(j)$ satisfying (2.1), degenerate functions
$f_s(x_1,\dots,x_s)$, $s=1,\dots,k$, and iid.\ random variables $X_1$,
$X_2$,\dots with uniform distribution in $[0,1]$. The joint distribution
of the
random variables $n^{-s/2}U_n^{(s)}$, $1\le s\le k$, tends to that of the
random vector
$$
V^{(s)}=\frac 1{k!}E^{s/2}\int
f_s(x_1,\dots,x_s)\,W(\,dx_1)\dots\,W(\,dx_s)\;,\quad 1\le s\le k\;,
$$
as $n\to\infty$, where $W(x)$ is a Wiener process on $[0,1]$.
\endproclaim
\demo{Proof of Theorem $3`$} The proof goes on the same line as that of
Theorem~3, only we need a multidimensional version of Lemmas~1, 2
and~3. We only explain the modified Lemmas we need during the proof. The
proof of Lemma~3 also yields that if the functions $f_s(x_1,\dots,x_s)$,
$1\le s\le k$ are square integrable and $\int h^2(y)\,dy=1$, then the
joint distribution of the vectors
$$
\align
I_1^{(s)}&=\int f_s(x_1,\dots,x_s)\,W(\,dx_1)\dots\,W(\,dx_s)\,,\quad1\le
s\le k\;,\\
\intertext{and}
I_2^{(s)}&=\int f_s(x_1,\dots,x_s)h(y_1)\cdots h(y_s)
\,B(\,dx_1,\,dy_1)\dots\,B(\,dx_s,\,dy_s)\,,\quad 1\le s\le k\;,
\endalign
$$
agree.
 
We need a multidimensional version of Lemma ~1, where we have to
approximate the sums
$$
U_n^{(s)}=\sum_{1\le<j_1<j_2\dots<j_s\le n}
a_{s}(j_1,\dots,j_s)f_s(X_{j_1},\dots,X_{j_s})\,\quad 1\le s\le k\;,
$$
simultaneously if  the functions $f_s$ satisfy $(1.4')$, and the
sequences $a_s(j_1,\dots,j_s)$ are all $\e$-approximable by a set of
elementary functions. We want to  get the same approximation of the
random variables $U_n^{(s)}$ as in Lemma~1 for all  $1\le s\le k$ (by
replacing $k$ by $s$ everywhere) with the following additional
restriction: The approximating sums must be the polynomials of the
same independent centered Poissonian random variables $\eta_{j,l}$ for
all $1\le s\le k$. This is possible if the following conditions are
satisfied. The $\e$-approximation of the function $a_s$ is determined at
level $n$ by a partition $\Lambda_1,\dots,\Lambda_p$ of $\{1,\dots,n\}$
independent of~$s$ together with some function
$B_{n,s}^{\e}(m_1,\dots,m_p)$, and the functions $f_s$ are
$\e$-approximated by such step functions $g^\e_s(x_1,\dots,x_k)$
which satisfy Remark~1 with the same constant $L$ in it for all $1\le
s\le k$. These conditions can be satisfied. If the
$\e$-approximation of the function $a_s$ is determined at level $n$ by a
partition $\Cal L_s=\{\Lambda_1(s),\dots,\Lambda_{p(s)}(s)\}$ of
$\{1,\dots,n\}$ which depends on $s$ and some function $B_{n,s}^\e$,
$1\le s\le k$, then it is also determined by a partition which is a
refinement of all partitions
$\Cal L_s$, $1\le s \le k$, and a function $B^\e_{n,s}$ such that
relation (3.2) remains valid on the new partition with the same function
$b_{n}^\e(j_1,\dots,j_s)=b_{n,s}^\e(j_1,\dots,j_s)$. To see that the
conditions of Remark~1 can be satisfied simultaneously for all
$f_s$, $1\le s\le k$, observe first that the functions $f_s$ can be well
approximated in $L^2$ norm by continuous functions. This implies that
Remark ~1 can be satisfied for all sufficiently large $L$. Then the
proof of Lemma~1 can be carried out to supply the strengthened form
of Lemma~1 needed for us.
 
Finally we need the following modified version of Lemma~2. In Lemma ~2
we took a polynomial of order $k$ of independent Gaussian and centered
Poissonian random variables, and showed that their characteristic
functions are close to each other under certain conditions. Take the
polynomials of order $s$ for all $1\le s\le k$ of the same random
variables, and assume that these polynomials satisfy the conditions of
Lemma~2. Consider the random vectors which we get when the
centered Poissonian and when the Gaussian random variables are chosen as
the arguments of these polymomials. Then the characteristic functions of
these random vectors are close to each other. This statement can be
proved in the same way as Lemma~2, and Theorem ~$3'$ can be proved by
means of these generalized lemmas just as Theorem~3.\qed
\enddemo
\proclaim{Lemma 4}\it Let the function $B_n(j_1,\dots,j_s)$, $1\le s\le
k$, be defined by (2.3) or $(\text{2.3}')$ with a function of the form
$a(j_1,\dots,j_k)=e(j_1)\cdots e(j_k)$. Assume that $e(j)$, $j=1$,
2,\dots, is a bounded sequence  satisfying (2.1) and such that
$\lim\limits_{n\to\infty} F_n=F$ for the sequence $F_n$ defined in
~(2.5). Then
$$
n^{-(k-s)/2}B_n(j_1,\dots,j_s)=\binom ks D_{k-s}e(j_1)\cdots
e(j_s)+\e_n^{(s)}(j_1,\dots,j_s) \tag4.7
$$
such that
$$
\lim_{n\to\infty}\sup_{1\le s\le k} \sup_{1\le j_1,\dots,j_s\le n}
\e^{(s)}_n(j_1,\dots,j_s)=0\;, \tag$4.7'$
$$
and the sequence $D_s$ is defined by the recursive formula $D_0=1$,
$D_1=F$, and
$$
D_s=F^s-\sum_{p=1}^{\[\frac s2\]} \frac{s!}{2^pp!(s-2p)!}E^p D_{s-2p}\;.
\tag4.8
$$
\endproclaim
\demo{Proof of Lemma 4} By formula $(2.3')$
$$
B_n(j_1,\dots,j_s)=\binom k s G_n(j_1,\dots,j_s)e(j_1)\cdots
e(j_s)\tag4.9
$$
with
$$
G_n(j_1,\dots,j_s)=\sum\Sb
l_p\in\{1,\dots,n\}\setminus\{j_1,\dots,j_s\},\;1\le p\le k-s\\
l_p\ne l_{p'}\text{ if }p\ne p'\endSb
e(l_1)\cdots e(l_{k-s})\;. \tag$4.9'$
$$
We need a good asymptotics for the term $G_n$ defined in $(4.9')$. For
this aim we introduce some notations. Given a finite set $A$ let $|A|$
denote its cardinality. For a set $U\subset \{1,\dots,k\}$ let $\Cal U_U$
denote the set of all partitions of the set $U$, and for a set
$J\subset\{1,\dots,n\}$ and a partition $(V_1,\dots,V_p)$ of
$U\subset\{1,\dots,k\}$ put
$$
H_{U,J}^{(n)}(V_1,\dots,V_p)=\sum\Sb j_s\in\{1,\dots,n\}\setminus
J,\;s\in U\\
j_s=j_{s'}\text{ if }j_s\in V_r,\;j_{s'}\in V_r \text{ for the same
}1\le r\le p\\
j_s\ne j_{s'}\text{ if }j_s\in V_r,\;j_{s'}\in V_{r'}\text{
for }r\ne r' \endSb
\prod_{s\in U} e(j_s)\;.
$$
Let us observe that $H_{U,J}^n(V_1,\dots,V_p)$ depends only on the
cardinalities $|V_1|,\dots, |V_p|$ but not on the exact form of the
sets
$V_1$,\dots, $V_p$. We claim that  if $|J|\le K$ with some fixed $K>0$,
then
$$
|H_{U,J}^{(n)}(V_1,\dots,V_p)|<\const n^{|U|/2} \tag4.10
$$
and
$$
|H_{U,J}^{(n)}(V_1,\dots,V_p)|<\const n^{(|U|-1)/2}\quad\text{if
}|V_r|\ge3 \text{ for some }1\le r\le p\;.\tag4.11
$$
We prove (4.10) by induction for the number of elements of the
partitions. It holds if the partition consists only of one elements,
since
$$
\left|\sum_{j\in\{1,\dots,n\}\setminus J} e(j)^{|U|}\right|<
\cases &\const \sqrt n\quad \text{if }|U|=1\\
 &\const  n\quad \text{if }|U|\ge2
\endcases \tag4.12
$$
Then relation (4.10) follows from the inductive hypothesis and the
identity
$$
\aligned
H_{U,J}^{(n)}(V_1,\dots,V_p)=& H_{U\setminus
V_1,J}^{(n)}(V_2,\dots,V_p)\sum_{j_s\in\{1,\dots,n\}\setminus
J \;\text{for }s\in V_1} e(j_s)^{|V_1|} \\
&\qquad-\sum_{i=2}^p
H_{U,J}^{(n)}(V_2,\dots,V_1\cup V_i,\dots,V_p)\;.
\endaligned \tag4.13
$$
It is enough to prove (4.11) in the case when $|V_1|\ge 3$. We can prove
it similarly to the relation (4.10) by induction for the number of
elements of the partition. If the partition consists of one element,
then (4.11) holds because of (4.12), and if it contains more than
one element, then it follows from the inductive hypothesis, (4.13),
(4.12) and (4.10).
 
To investigate those partitions of a set $U$ which consist of sets with
cardinality one or two we introduce the quantities: $$
\align
H^{(n)}_J(r,s)=H_{U,J}^{(n)}&(\{1,2\},\dots,\{2r-1,2r\},\{2r+1\},
\dots,\{2r+s\})\\
&\qquad\text{ with }U=\{1,\dots,2r+s\}\;.
\endalign
$$
For $J=\emptyset$ put
$$
H^{(n)}(r,s)=H^{(n)}_\emptyset(r,s)\;.
$$
We claim that
$$
\left|H^{(n)}_J(r,s)-n^{r}E_n^r H^{(n)}(0,s)\right|<\const
n^{(2r+s-1)/2}\tag4.14
$$
if $|J|\le K$ with some $K>0$, where
$E_n=\dfrac1n\sum\limits_{j=1}^n e(j)^2$. To prove (4.14) observe that
$$
n^{r}E_n^r H^{(n)}(0,s)=\sum\Sb j_u\in\{1,\dots,n\}\text { for
}1\le u\le 2r+s\\
j_{2u-1}=j_{2u}\text{ for }1\le u\le r\\
j_u\ne j_{u'}\text{ if }2r< u,\,u'\le 2r+s\text{ and }u\ne u'\endSb
e(j_1)\cdots e(j_{2r+s})\;.                             \tag4.15
$$
Hence
$$
\left|H^{(n)}_J(r,s)-n^{r}E_n^s
H^{(n)}(0,s)\right|\le\Sigma_1+\Sigma_2
$$
with
$$
\Sigma_1=\[\sup_{1\le j\le n} |e(j)|^{2r+s}+1\] \((2r+s)|J|\)^{2r+s}\sum
\Sb |U|\le 2r+s-1\\
(V_1,\dots,V_p)\in \Cal U_U\endSb
|H^{(n)}_{U,J}(V_1,\dots,V_p)|
$$
and
$$
\Sigma_2=\sum_{(V_1,\dots,V_p)\in\Cal
V}|H^{(n)}_{V,J}(V_1,\dots,V_p)|\;, $$
where $V=\{1,\dots,2r+s\}$, and $\Cal V$ denotes the set of those
partitions of $V$ whose elements are unions of the sets $\{1,2\}$,\dots,
$\{2r-1,2r\}$, $\{2r+1\}$,\dots, $\{2r+s\}$ and it contains at least
one
set such that it has a proper subset of the form $\{2j-1,2j\}$, $1\le
j\le r$.
Here $\Sigma_1$ bounds the contribution of those products $e(j_1)\cdots
e(j_{2r+s})$ in (4.15) which contain a term $e(j_l)$ with $j_l\in J$,
and $\Sigma_2$ bounds the contribution of those products for which
$e(j_l)\in\{1,\dots,n\}\setminus J$ for all $1\le l\le 2r+s$, but do not
appear in the expression defining $H^{(n)}_{J,V}(r,s)$. The relations
$\Sigma_1\le\const n^{(2r+s-1)/2}$ and
$\Sigma_2\le\const n^{(2r+s-1)/2}$ hold because of formulas (4.12) and
(4.13) respectively.
 
We shall prove by induction  for $s$ that
$$
\lim_{n\to \infty} n^{-s/2} H^{(n)}(0,s)=D_s \tag4.16
$$
with the sequence $D_s$ defined in (4.8). Indeed, (4.16) holds for $s=1$
and for $s \ge2$ we can write
$$
H^{(n)}(s)=n^{s/2}F_n^s-\sum_{(V_1,\dots,V_p)\in \Cal
U_s\setminus(\{1\},\dots,\{s\})} H^{(n)}_{U,\emptyset}(V_1,\dots,V_p)
$$
where $\Cal U_s$ denotes the set of partitions of $U=\{1,\dots,s\}$. We
get relation (4.16) by dividing in the last relation by $n^{-s/2}$ and
taking
limit $n\to\infty$ if we use relations (4.11), (4.14), the induction
hypothesis, the relation $\lim\limits_{n\to\infty}E_n=E$,
$\lim\limits_{n\to\infty}F_n=F$ and the fact
that the set $\{1,\dots,s\}$
contains $\dfrac{s!}{2^pp!(s-2p)!}$ partitions consisting of $p$ sets
with cardinality~2 and $s-2p$ sets with cardinality~1, \ $1<2p\le s$.
 
Clearly, for the expression $G_n$ defined in $(4.9')$
$G_n(j_1,\dots,j_s)= H_J(0,k-s)$ with $J=\{j_1,\dots,j_s\}$. Hence
relations (4.16) and (4.14) imply that
$$
\lim_{n\to\infty}n^{-(k-s)/2}G_n(j_1,\dots,j_s)=D_{k-s}
$$
and the convergence is uniform in  $(j_1,\dots,j_s)$. The last relation
together with formula (4.9) imply Lemma~4. \qed
\enddemo
\demo{Proof of Theorem 4} We get by rewriting the expression (2.4) by
means of the Hoeffding decomposition, and applying Lemma~4 that
$$
n^{-k/2}U_n=V_n+\eta_n
$$
with
$$
V_n=\sum_{s=1}^k n^{-s/2}\binom ks\frac1{k!}D_{k-s}\sum\Sb 1\le
j_p\le n,\text { for }1\le p\le s\\ j_p\ne j_{p'}\text{ if }p\ne
p'\endSb e(j_1)\cdots e(j_s) f_s(X_{j_1},\dots, X_{j_s})
$$
and
$$
\eta_n=\sum_{s=1}^k n^{-s/2} \frac1{k!}\sum\Sb 1\le
j_p\le n,\text { for }1\le p\le s\\ j_p\ne j_{p'}\text{ if }p\ne
p'\endSb  \e_n^{(s)}(j_1,\dots,j_s) f_s(X_{j_1},\dots, X_{j_s})\;.
$$
The random variables $f(X_{j_1},\dots,X_{j_s})$ and
$f(X_{j'_1},\dots,X_{j'_s})$ are uncorrelated if the sets %@j'_s esnemj,_s
$\{j_1,\dots,j_s\}$ and $\{j'_1,\dots,j'_s\}$ are different, since the
functions $f_s$ satisfy relation $(1.4')$. Hence formula $(4.7')$
implies that $E\eta_n^2\to 0$ as $n\to\infty$, and $n^{-1/2}U_n$ and
$V_n$ have the same limit distribution as $n\to \infty$. By Theorem
$3'$ the random variables $V_n$ have the
limit distribution given in Theorem~4. \qed
\enddemo
\demo{Remark 2} If $\lim\limits_{n\to\infty} F_n=\infty$, and the
remaining conditions of Theorem~4 hold and $s$ is the
smallest index such that the function $f_s$ in (1.4) does not vanish
identically, then the sequence $n^{-k/2}F_n^{s-k}U_n$ converges in
distribution to the stochastic integral
$$
\frac{E^{s/2}}{s!(k-s)!}\int
f_s(x_1,\dots,x_s)\,W(\,dx_1)\dots\,W(\,dx_s)
$$
as $n\to\infty$. This can be proved similarly to Theorem~4, the only
difference is that now the behavior of the coefficint $B_n$ defined in
(2.3) is different. In this case
$$
B_n(j_1,\dots,j_s)\approx n^{(k-s)/2} F_n^{k-s}e(j_1)\cdots e(j_s)\;.
$$
The problem can be handled similarly in the case when
$\lim\limits_{n\to\infty} F_n=0$. Here again a good asymptotics is
needed for the function $B_n$. In this case the great indices $s$ count
for which the function $f_s$ does not vanish in the Hoeffding
decomposition (1.4). But the situation is more complicated in this case.
The asymptotic behavior of the sums $\sum\limits_{j=1}^ne(j)^r$ can play
a role not only for $r=1$ or~2. We omit a closer investigation of this
problem. \enddemo\bigskip
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\parindent=17pt \smallskip
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\bigskip\bigskip\rightline{\vbox{\halign{\smc #\hfill\cr
Mathematical Institute of the\cr
\ \ Hungarian Academy of Sciences\cr
P.O.B.\ 127\cr
Budapest H-1364\cr
Hungary\cr}}}
\bye