\magnification=\magstep1
\input amstex
\documentstyle{amsppt}
\TagsOnRight
\nopagenumbers
\hsize=15truecm
\parskip=3pt
\define\({\left(}
\define\){\right)}
\define\e{\varepsilon}
\define\[{\left[}
\define\]{\right]}
\define\const{\text{\rm const.}\,}
\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:
$$
U_n=\sum_{1\le j_1x)
1
$$
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
$$
B_n(j_1,\dots,j_s)=\sum\Sb 1\le l_p\le n,\; 1\le p\le k\\
l_p\ne l_{p'}\text{ if }p\ne p'\\
j_p=l_{r_p} \text{ for some }r_p\quad p=1,\dots,s\\
\text{such that }1\le r_1<\cdots0$ an
approximating step-function
$g(x_1,\dots,x_k)=g_\e(x_1,\dots,x_k)$ can be given such that
$$
\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)|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}Tn(\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
$$
and the sequence $\bar e(j)$, $j=1$, 2,\dots takes finitely many values
$K_1\e0$,
then
$$
|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
\centerline{REFERENCES}
\parindent=17pt \smallskip
\item{[1]} Dynkin, E. B., Mandelbaum, A. (1983). Symmetric statistics,
Poisson processes and multiple Wiener integrals. {\it Annals of
Statistics\/} {\bf 11} 739--745
\item{[2]} Hal\'asz, G., Major, P. (1977). Reconstructing the
distribution from partial sums of samples. {\it The Annals of Probability\/}
{\bf 5} 987--998
\item{[3]}It\^o, K. (1951). Multiple Wiener integral. {\it J.
Math.\ Soc.\ Japan\/} {\bf 3} 157--164
\item{[4]} Janson, S. (1984). The asymptotic distribution of incomplete
$U$-statistics. {\it Z. Wahrscheinlichkeitstheorie verw.
Gebiete\/} {\bf 66} 495--505
\item{[5]} Koml\'os, J., Major, P., Tusn\'ady, G. (1975). An
approximation of partial sums of independent RV's and the sample DF.~I
{\it Z. Wahrscheinlichkeitstheorie verw.\ Gebiete\/} {\bf 32} 111--131
\item{[6]} Lieb, E. H., Lebowitz, J. L. (1972). The constitution of
matter: existence of thermodynamics for systems composed of electrons
and nuclei. {\it Advances in Mathematics\/} {\bf 9} 316--318
\item{[7]} Major, P. (1981). Multiple Wiener--It\^o integrals. {\it
Lecture Notes in Mathematics\/} {\bf 849}, Springer--Verlag, Berlin
Heidelberg New York
\item{[8]} Mc.\ Kean, H. P. (1973). Geometry of differential spaces.
{\it Annals of Probability\/} {\bf 1} 197--206
\item{[9]} O'Neil, K. A., Redner, R. A. (1992). Asymptotic
distributions of weighted $U$-statistics of degree two. {\it Annals of
Probability\/} (to appear)
\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