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\centerline{\bf An estimate on the supremum of a nice class of}
\centerline{\bf stochastic integrals and $U$-statistics.}
\smallskip
\centerline{\it P\'eter Major}
\centerline{Alfr\'ed R\'enyi Mathematical Institute of the Hungarian
Academy of Sciences}
\centerline{Budapest, P.O.B. 127 H--1364, Hungary, e-mail:
major\@renyi.hu}
\medskip
{\narrower \noindent {\it Summary:}\/
Let a sequence of iid. random variables $\xi_1,\dots,\xi_n$ be
given on a space $(X,\Cal X)$ with distribution $\mu$ together
with a nice class $\Cal F$ of functions $f(x_1,\dots,x_k)$ of $k$
variables on the product space  $(X^k,\Cal X^k)$. For all
$f\in \Cal F$ we consider the random integral  $J_{n,k}(f)$ of the
function $f$ with respect to the $k$-fold product of the normalized
signed measure $\sqrt n(\mu_n-\mu)$, where $\mu_n$ denotes the
empirical measure defined by the random variables
$\xi_1,\dots,\xi_n$ and investigate the probabilities
$P\(\supp_{f\in \Cal F}|J_{n,k}(f)|>x\)$ for all $x>0$. We show
that for nice classes of functions, for instance if $\Cal F$ is a
Vapnik--\v{C}ervonenkis class, an almost as good bound can be given
for these probabilities as in the case when only the random integral
of one function is considered. A similar result holds for degenerate
$U$-statistics, too. \par}
 
\beginsection 1. Introduction. Formulation of the main results
 
In some investigations about non-parametric maximum likelihood
estimates (see~[10] or~[11]) I met the problem how to give a good
estimate about the distribution of the supremum of appropriate
classes of multiple integrals with respect to a normalized
empirical measure. This problem is closely related to the study of
the supremum of good classes of degenerate $U$-statistics. Hence, it
is natural to study these two problems simultaneously. This will be
done in the present paper. To formulate its main results
first I introduce some notations and recall some definitions.
 
Let a probability measure $\mu$ be given on a measurable space
$(X,\Cal X)$, take a sequence $\xi_1,\dots,\xi_n$ of independent,
identically distributed $(X,\Cal X)$ valued random variables with
distribution~$\mu$, and define the empirical measure $\mu_n$,
$$
\mu_n(A)=\dfrac1n\#\{j\colon\xi_j\in A,\;1\le j\le n\},\quad A\in\Cal X,
\tag1.1
$$
of the sample $\xi_1,\dots,\xi_n$. Let us take a nice set $\Cal F$
of measurable functions $f(x_1,\dots,x_k)$ on the $k$-fold product
space $(X^k,\Cal X^k)$ and define the integrals $J_{n,k}(f)$ of
the functions $f\in \Cal F$ with respect to the $k$-fold product
of the normalized empirical measure $\sqrt n(\mu_n-\mu)$ by the
formula
$$
\align
J_{n,k}(f)&=\dfrac{n^{k/2}}{k!} \int'
f(x_1,\dots,x_k)(\mu_n(\,dx_1)-\mu(\,dx_1))\dots
(\mu_n(\,dx_k)-\mu(\,dx_k)),\\
&\qquad\text{where the prime in $\tsize\int'$ means that the
diagonals } x_j=x_l,\; 1\le j<l\le k,\\
&\qquad\text{are omitted from the domain of integration.} \tag1.2
\endalign
$$
I try to give a good estimate on the probabilities
$P\(\supp_{f\in\Cal F}|J_{n,k}(f)|>x\)$ for all $x>0$. To formulate
the result in this direction first I introduce the following definition.
\medskip\noindent
{\bf Definition of $L_p$-dense classes of functions.} {\it Let us
have a measurable space $(Y,\Cal Y)$ and a set $\Cal G$ of
$\Cal Y$-measurable functions on this space. We call $\Cal G$ an
$L_p$-dense class with parameter $D$ and exponent $L$ if for all
numbers $1\ge\e>0$ and probability measures $\nu$ on the space
$(Y,\Cal Y)$ there exists a finite $\e$-dense subset
$\Cal G_{\e,\nu}=\{g_1,\dots,g_m\} \subset \Cal G$ in the space
$L_p(Y,\Cal Y,\nu)$ consisting of $m\le D\e^{-L}$ elements, i.e.\
there is such a set $\Cal G_{\e,\nu}\subset \Cal G$ for which
$\inff_{g_j\in \Cal G_{\e,\nu}}\int |g-g_j|^p\,d\nu<\e^p$
for all functions $g\in \Cal G$. (Here the set $\Cal G_{\e,\nu}$ may
depend on the measure $\nu$, but its cardinality is bounded by a
number depending only on $\e$.)}
\medskip
In this paper we shall work with such classes of functions $\Cal F$
which contain only functions with absolute value less than or equal
to~1. In this case $\Cal F$ is an $L_p$-dense class of functions for
all $1\le p<\infty$ (with an exponent and a parameter depending
on~$p$) if there is a number $1\le p<\infty$ for which it is
$L_p$-dense. We shall formulate our statements mainly for $L_p$-dense
classes of functions with the parameter $p=2$, since this seems to be
the most convenient choice. Our main result is the following
\medskip\noindent
{\bf Theorem 1.} {\it Let us have a non-atomic measure $\mu$ on the
space $(X,\Cal X)$ together with an $L_2$-dense class $\Cal F$ of
functions $f=f(x_1,\dots,x_k)$ of $k$ variables with some parameter
$D>0$ and exponent $L\ge1$ on the product space $(X^k,\Cal X^k)$ which
consists of at most countably infinite functions, and satisfies the
conditions
$$
\|f\|_\infty=\supp_{x_j\in X,\;1\le j\le k}|f(x_1,\dots,x_k)|\le 1,
\qquad \text{for all } f\in \Cal F \tag1.3
$$
and
$$
\|f\|_2^2=Ef^2(\xi_1,\dots,\xi_k)=\int f^2(x_1,\dots,x_k)
\mu(\,dx_1)\dots\mu(\,dx_k)\le \sigma^2 \qquad \text{for all }
f\in \Cal F \tag1.4
$$
with some constant $0<\sigma\le1$.
Then there exist some constants $C=C(k)>0$, $\alpha=\alpha(k)>0$
and $M=M(k)>0$ depending only on the parameter $k$ such that the
supremum of the random integrals $J_{n,k}(f)$, $f\in \Cal F$,
defined by formula (1.2) satisfies the inequality
$$
\aligned
P\(\supp_{f\in\Cal F}|J_{n,k}(f)|\ge x\)&\le CD \exp\left\{-\alpha
\(\frac x{\sigma}\)^{2/k}\right\} \\
&\qquad \text{if}\quad n\sigma^2\ge
\(\frac x\sigma\)^{2/k} \ge M(L+\beta)^{3/2}\log\frac2\sigma,
\endaligned \tag1.5
$$
where $\beta=\max\(\frac{\log D}{\log n},0\)$,
and the
numbers $D$ and $L$ in formula~(1.5) are the parameter and exponent
of the $L_2$-dense class~$\Cal F$.}
\medskip
 
Theorem 1 has a natural counterpart about degenerate $U$-statistics
formulated in Theorem~2 below. Before its formulation I recall the
definition of $U$-statistics and degenerate $U$-statistics.
 
Let us have a sequence of independent and identically distributed
random variables $\xi_1,\xi_2,\dots$ with distribution~$\mu$ on a
measurable space $(X,\Cal X)$ together with a function
$f=f(x_1,\dots,x_k)$ on the $k$-th power $(X^k,\Cal X^k)$ of the
space $(X,\Cal X)$. We define with their help the
$U$-statistic~$I_{n,k}(f)$ of order~$k$, as
$$
I_{n,k}(f)=\frac1{k!}\summ\Sb 1\le j_s\le n,\; s=1,\dots, k\\
j_s\neq j_{s'} \text{ if } s\neq s'\endSb
f\(\xi_{j_1},\dots,\xi_{j_k}\).   \tag1.6
$$
(The function $f$ in this formula will be called the kernel
function of the $U$-statistic.)
 
A real valued function $f=f(x_1,\dots,x_k)$ on the $k$-th power
$(X^k,\Cal X^k)$ of a space $(X,\Cal X)$ is called a canonical
kernel function (with respect to the probability measure $\mu$
on the space $(X,\Cal X)$) if
$$
\int f(x_1,\dots,x_{j-1},u,x_{j+1},\dots,x_k)\mu(\,du)=0\quad
\text{for all } 1\le j\le k \text{ \ and \ } x_s\in X,  \; s\neq j.
$$
I also introduce the notion of canonical functions in a more
general case, because this notion appears later in Proposition~5 of
this paper. We call a function $f(x_1,\dots,x_k)$ on the $k$-fold
product $(X_1\times\cdots\times X_k, \Cal X_1\times\cdots\times
\Cal X_k, \mu_1\times\cdots\times \mu_k)$ of $k$ not necessarily
identical probability spaces $(X_j,\Cal X_j,\mu_j)$, $1\le j\le k$,
canonical if
$$
\int f(x_1,\dots,x_{j-1},u,x_{j+1},\dots,x_k)\mu_j(\,du)=0\quad
\text{for all }1\le j\le k \text{ \ and \ } x_s\in X_s,\; s\neq j.
$$
A $U$-statistic with a canonical kernel function is called degenerate.
Now I formulate Theorem~2.
\medskip\noindent
{\bf Theorem 2.} {\it Let us have a probability measure $\mu$ on
a space $(X,\Cal X)$, a sequence of independent and $\mu$
distributed random variables $\xi_1,\dots,\xi_n$ together with an
$L_2$-dense class $\Cal F$ of canonical (with respect to the
measure~$\mu$) kernel functions $f=f(x_1,\dots,x_k)$ with some
parameter $D>0$ and exponent $L\ge1$ on the product space
$(X^k,\Cal X^k)$ which consists of at most countably infinite functions,
and satisfies conditions (1.3) and (1.4) with some $0<\sigma\le1$.
Then there exist some numbers $C=C(k)>0$, $M=M(k)>0$
$\alpha=\alpha(k)>0$ depending only on the order $k$ of the
$U$-statistics we consider such that the degenerate
$U$-statistics $I_{n,k}(f)$, $f\in\Cal F$, defined in (1.6)
satisfy the inequality
$$
\aligned
P\(\supp_{f\in\Cal F}|n^{-k/2}I_{n,k}(f)|\ge x\)&\le CD
\exp\left\{-\alpha \(\frac x{\sigma}\)^{2/k}\right\} \\
&\qquad \text{if}\quad n\sigma^2\ge
\(\frac x\sigma\)^{2/k} \ge M(L+\beta)^{3/2}\log\frac2\sigma,
\endaligned \tag1.7
$$
where $\beta=\max\(\frac{\log D}{\log n},0\)$.}
\medskip
To understand the relation between Theorems~1 and~2 let us observe
that the definition of the $U$-statistics in (1.6) can be rewritten as
$$
I_{n,k}(f)=\frac{n^k}{k!}\int' f(x_1,\dots,x_k)\mu_n(\,dx_1)\dots
\mu_n(\,dx_k) \tag1.8
$$
if the distribution $\mu$ of the random variables $\xi_1,\dots,\xi_n$
is non-atomic. (The non-atomic property is needed to guarantee that
the random variables $\xi_1,\dots,\xi_n$ take different values with
probability~1.) The difference between the random integrals $J_{n,k}(f)$
and the random integral representation~(1.8) of $U$-statistics
$I_{n,k}(f)$ is (beside the different norming constant) that in
formula~(1.8) we integrate with respect to the empirical measure
$\mu_n$ and not with respect to its normalized version $\mu_n-\mu$.
As a consequence, we can get a good estimate for $U$-statistics
only under some restriction. In Theorem~2 we had to impose the
condition that the functions of the class $\Cal F$ are canonical,
while no similar condition was needed in Theorem~1. Hence
Theorem~1 can be better applied in statistical problems. On the
other hand, the proof of Theorem~2 is simpler. But Theorem~1 can
be deduced from it by means of a good representation of multiple
random integrals $J_{n,k}(f)$ as a linear combination of degenerate
$U$-statistics. In this work the following approach will be
followed. In the main text Theorem~2 will be proved. The Appendix
contains the proof of the above mentioned representation of random
integrals which enables us to deduce Theorem~1 from Theorem~2.
 
Let us discuss the conditions of Theorems~1 and~2. We have assumed
that $\Cal F$ contains at most countably infinite functions.
This condition, which is too restrictive for statistical
applications can be weakened. The introduction of the following
definition seems to be useful.
\medskip\noindent
{\bf Definition of countable approximability.} {\it A class of
functions $\Cal F$ is countably approximable in the space
$(X^k,\Cal X^k,\mu^k)$ if there exists a countable subset $\Cal
F'\subset \Cal F$ such that for all numbers $x>0$ the sets
$A(x)=\{\oo\colon\supp_{f\in \Cal F}|J_{n,k}(f)(\oo)|\ge x\}$ and
$B(x)=\{\oo\colon\supp_{f\in \Cal F'}|J_{n,k}(f)(\oo)|\ge x\}$ satisfy
the identity $P(A(x)\setminus B(x))=0$.}
\medskip
Clearly, $B(x)\subset A(x)$. In the above definition we demanded
that for all $x>0$ the set $B(x)$ is almost as large as $A(x)$.
The following corollary of Theorems~1 and~2 holds.
\medskip\noindent
{\bf Corollary of Theorem~1 or~2.} {\it Let a class of functions
$\Cal F$ satisfy the conditions of Theorem~1 or~2 with the only
exception that instead of the condition about the countable
cardinality of $\Cal F$ it is assumed that $\Cal F$ is countably
approximable in the space $(X^k,\Cal X^k,\mu^k)$. Then $\Cal F$
satisfies Theorem~1 or~2.}
\medskip
In Theorems~1 and~2 we have imposed the condition that the class of
functions $\Cal F$ is countable to avoid some unpleasant measure
theoretical difficulties. Otherwise we should have to work with
possibly non-measurable sets. On the other hand, I have the
impression that Corollary~1 can be applied in all statistical
problems where we have to work with the supremum of multiple
random integrals or $U$-statistics. It is not difficult to prove
that Corollary~1 follows from Theorem~1 or~2. We have to show
that if $\Cal F$ is an $L_2$-dense class with some parameter $D$
and exponent $L$, and $\Cal F'\subset \Cal F$, then $\Cal F'$ is
also an $L_2$-dense class with the same exponent $L$, only with a
possibly different parameter~$D'$.
 
To prove this statement let us choose for all numbers $1\ge\e>0$
and probability measures $\nu$ on $(Y,\Cal Y)$ some functions
$f_1,\dots,f_m\in \Cal F$ with $m\le D\(\frac\e2\)^{-L}$ elements,
such that the sets $\Cal U_j=\left\{f\colon\int |f-f_j|^2\,d\nu\le
\(\frac\e2\)^2\right\}$ satisfy the relation $\bigcupp_{j=1}^m
\Cal U_j=Y$. For all sets $\Cal U_j$ for which $\Cal U_j\cap
\Cal F'$ is non-empty choose a function $f'_j\in \Cal U_j\cap
\Cal F'$. In such a way we get a collection of functions $f'_j$
from the class $\Cal F'$ containing at most $2^LD\e^{-L}$ elements
which satisfy the condition imposed for $L_2$-dense classes with
exponent $L$ and parameter $2^LD$ for this number $\e$ and measure
$\nu$.
 
In Theorems~1 and~2 we have considered the supremum of multiple
random integrals and $U$-statistics of order~$k$ for a nice class
of functions. It was shown that if the variances of the random
integrals or $U$-statistics we have considered are less than some
number $0<\sigma^2\le1$, (formula (1.4) was a condition about these
variances in an implicit way) then under some additional conditions
this supremum takes a value larger than $x$ with a probability less
than $P(C\sigma\eta^k>x)$, where $\eta$ is a standard normal
random variable, and $C=C(k)>0$ is a universal constant depending
only on the multiplicity $k$ of the random integrals. This is the
sharpest estimate we can expect. Moreover, this estimate seems to
be sharp also in that respect that the conditions imposed for its
validity cannot be considerably weakened. If condition (1.3) does
not hold or $n\sigma^2<\(\frac x\sigma\)^{2/k}$, then the estimate
of Theorem~1 or~2 may not hold any longer even if the class of
functions $\Cal F$ contains only one function. In such cases there
exist examples for which the probability $P(J_{n,k}(f)>x)$ is too
large. In~[8] I gave such examples (Examples~3.2 and~8.6).
Here I do not discuss them in detail.
 
If the other inequality is violated in the conditions of
formula~(1.5) or~(1.7), i.e. if $\(\frac x\sigma\)^{2/k}
<M\log\frac2\sigma$ with a not too large number $M>0$, then the
estimate of Theorem~1 or~2 may not hold for a different reason.
The supremum of many small random variables may be large, and
inequalities~(1.5) or~(1.7) may loose their validity for this
reason. To understand this let us consider the following analogous
problem. Take a Wiener process $W(t)$, $0\le t\le1$, and consider
the supremum of the expressions $W(t)-W(s)=\int f_{s,t}(u)W(\,du)
=\bar J(f_{s,t})$, with the functions $f_{s,t}(\cdot)$ on the
interval $[0,1]$ defined by the formula $f_{s,t}(u)=1$ if $s\le
u\le t$, $f_{s,t}(u)=0$ if $0\le u<s$ or $t<u\le1$. If we consider
the class of functions $\Cal F_\sigma=\{f_{s,t}\colon\int
f^2_{s,t}(u)\,du=t-s\le\sigma^2\}$, then it is natural to expect
that $P\(\supp_{f_{s,t}\in\Cal F_\sigma} \bar J(f_{s,t})>x\)\le
e^{-\const (x/\sigma)^2}$. However, this relation does not hold if
$x=x(\sigma)<(1-\e)\sqrt{2\log\frac1\sigma}\sigma$ with some $\e>0$.
In such cases $P\(\supp_{f_{s,t}\in\Cal F_\sigma}\bar
J(f_{s,t})>x\)\to1$, as $\sigma\to0$. This can be proved relatively
simply with the help of the estimate $P(\bar J(f_{s,t})>x(\sigma))
\ge\const \sigma^{1-\e}$ if $|t-s|=\sigma^2$ and the independence of
the random integrals $\bar J(f_{s,t})$ if the functions $f_{s,t}$ are
indexed by such pairs $(s,t)$ for which the intervals $(s,t)$ are
disjoint.
 
Some additional work would show that a similar picture arises if
we integrate with respect to the normalized empirical measure of a
sample with uniform distribution on the interval $[0,1]$ instead
of a Wiener process. This yields an example for an $L_2$-dense
class of functions in the case $k=1$ for which the estimate of
Theorem~1 does not hold any longer if $\(\frac x\sigma\)^{2/k}
<M\log\frac2\sigma$ with some $M<\sqrt2$. Similar example can be
constructed also in the case of Theorem~2. At a heuristic level
it is clear that such an example can be given also for $k>1$, and
the number $M$ in condition~(1.5) or~(1.7) has to be chosen larger
if we want that Theorem~1 or Theorem~2 hold also for an $L_2$-dense
class of functions $\Cal F$ with a large exponent~$L$. (In this
paper I did not try to find the best possible condition of
Theorem~1 or~2 in the right-hand side inequality of~(1.5) or~(1.7).)
 
One would like to see some interesting examples when  Theorem~1 or~2
is applicable and to have some methods to check their conditions.
It is useful to know that if $\Cal F$ is a Vapnik--\v{C}ervonenkis
class of functions whose absolute values are bounded by 1, then
$\Cal F$ is an $L_2$-dense class.
 
To formulate the above statement more explicitly let us recall that
a class of subsets $\Cal D$ of a set $S$ is a Vapnik--\v{C}ervonenkis
class if there exist some constants $B>0$ and $K>0$ such that for
all integers $n$ and sets $S_0(n)=\{x_1,\dots,x_n\}\subset S$ of
cardinality $n$ the collection of sets of the form $S_0(n)\cap D$,
$D\in\Cal D$, contains no more than $Bn^K$ subsets of $S_0(n)$. A
class of real valued functions $\Cal F$ on a space $(Y,\Cal Y)$
is a Vapnik--\v{C}ervonenkis class if the graphs of these functions
is a Vapnik--\v{C}ervonenkis class, i.e.\ if the sets
$A(f)=\{(y,t)\colon y\in Y,\;\min(0,f(y))\le t\le\max(0,f(y))\}$,
$f\in \Cal F$, constitute a Vapnik--\v{C}ervonenkis class of sets on
the product space $Y\times R^1$.
 
An important result of Dudley states that a Vapnik--\v{C}ervonenkis
class of functions whose absolute values are bounded by 1 is an
$L_1$-dense class. The parameter and exponent of this $L_1$-dense
class can be bounded by means of the constants $B$ and $K$ appearing
in the definition of Vapnik--\v{C}ervonenkis classes. Beside this,
an $L_1$-dense class of functions bounded by 1 is also an
$L_2$-dense class (with possibly different exponent and parameter),
since $\int|f-g|^2\,d\nu\le2\int|f-g|\,d\nu$ in this case. Dudley's
result, whose proof can be found e.g.\ in Chapter~II of Pollard's
book~[9] (the 25$^\circ$ approximation lemma contains this result in
a slightly more general form) is useful for us, because there are
results which enable us to prove that certain classes of functions
constitute a Vapnik--\v{C}ervonenkis class.
 
I found some results similar to that of this paper in the work of
Arcones and Gine~[3], where the tail-behaviour of the supremum of
degenerate $U$-statistics was investigated if the kernel functions
of these $U$-statistics constitute a Vapnik--\v{C}ervonenkis class.
But the bounds of that paper do not give a better estimate if we
have the additional information that the variances of the
$U$-statistics we consider are small. The main goal of the present
paper was to prove such estimates which take into account the bound
we have on the variance of the random integrals $J_{n,k}(f)$ or
$U$-statistics $I_{n,k}(f)$ we consider.
 
In the investigation of this work Alexander's paper~[1] played an
essential role. In Alexander's work a similar problem was considered
in the special case $k=1$. It was interesting for me first
of all, because I learned some ideas from it which I strongly needed
in the present work. On the other hand, I also needed some new
arguments, because in the study of multiple stochastic integrals or
$U$-statistics some new difficulties had to be overcome.
 
This paper consists of six sections and an Appendix. In Section~2
Theorems~1 and~2 are reduced to two simpler statements formulated
in Propositions~2 and~3. Section~3 contains some important results
needed in the proof of Proposition~2, and the main ideas of its
proof are explained there. It is shown that Proposition~2 follows
from another statement formulated in Proposition~4. Proposition~4
is proved together with another result described in Proposition~5.
To make the proof more transparent first I explain it in the
special case $k=1$ in Section~4. Sections~5 and~6 contain the
proof of Propositions~4 and~5 in the general case. In Section~5
it is shown how a symmetrization argument can be applied to prove
Propositions~4 and~5, and finally the proof is completed in
Section~6. The Appendix contains the proof about a result of an
expansion of multiple random integrals in the form of a linear
combination of degenerate $U$-statistics formulated in Proposition~3.
This result enables us to deduce Theorem~1 from Theorem~2.
 
%\beginsection 2. Reduction of Theorems 1 and 2 to some simpler results
 
\medskip\noindent
{\bf 2. Reduction of Theorems 1 and 2 to some simpler results}
\medskip\noindent
First I prove with the help of a natural argument, called the
Chaining argument in the literature, and the multi-dimensional
generalization of Bernstein's inequality (see~[2],
Proposition~2.3(c)) a result that yields a reduction of
Theorem~2 we shall need later. I shall apply the following
consequence of this result (which is actually
equivalent to it).
 
If $U_{n,k}(f)$ is a degenerate $U$-statistic of order~$k$ with
a (canonical) kernel function $f$ which satisfies relations (1.3)
and (1.4) (formally the class of functions $\Cal F$ consisting only
of the function $f$ satisfies these relations) with some number
$0<\sigma\le1$ and the distribution $\mu$ of the iid. sequence
of the random variables $\xi_1,\dots,\xi_n$ taking part in the
definition of the $U$-statistic $U_{n,k}(f)$, then there exist
some constants $C=C(k)>0$ and $\alpha=\alpha(k)>0$ depending
only on the order $k$ of this~$U$-statistic such that
$$
P\(n^{-k/2}|I_{n,k}(f)|>x\)\le C \exp\left\{-\alpha
\(\frac x\sigma\)^{2/k}\right\} \quad\text{for } 0\le x\le
n^{k/2}\sigma^{k+1}.   \tag2.1
$$
Now I formulate the following result.
\medskip\noindent
{\bf Proposition 1.} {\it Let us fix some number $\bar A\ge2^k$,
and assume that a class of functions $\Cal F$ satisfies the
conditions of Theorem~2 with an appropriately chosen number $M$
in these conditions which may depend also on $\bar A$. Then a
number $\bar\sigma$, $0\le\bar\sigma\le \sigma\le1$, and a collection
of functions $\Cal F_{\bar\sigma}=\{f_1,\dots,f_m\}\subset \Cal F$
with $m\le D\bar\sigma^{-L}$ elements can be chosen in such a way
that the sets $\Cal D_j=\{f\colon f\in \Cal F,\int|f-f_j|^2\,d\mu
\le\bar\sigma^2\}$, $1\le j\le m$, satisfy the relation
$\bigcupp_{j=1}^m\Cal D_j=\Cal F$, and
$$ \allowdisplaybreaks
\align
P&\(\sup_{f\in\Cal F_{\bar\sigma}} n^{-k/2}|I_{n,k}(f)|\ge
\frac x{\bar A}\)\le 2CD\exp\left\{-\alpha
\(\frac x{10\bar A\sigma}\)^{2/k}\right\}  \tag2.2  \\
&\qquad \qquad \text{if}\quad n\sigma^2\ge
\(\frac x\sigma\)^{2/k} \ge ML\log\frac2\sigma
\endalign
$$
with the constants $\alpha=\alpha(k)$, $C=C(k)$ appearing in
formula~(2.1) and the exponent $L$ and parameter $D$ of the
$L_2$-dense class $\Cal F$ if the constant $M=M(k,\bar A)$ is
chosen sufficiently large. Beside this, also the inequalities
$4\(\frac x{\bar A\bar\sigma}\)^{2/k}\ge n\bar\sigma^2\ge
\frac1{64} \(\frac x{\bar A\sigma}\)^{2/k}$ and
$n\bar\sigma^2\ge\frac{M^{2/3}(L+\beta)\log n}{1000\bar A^{4/3}}$
hold, provided that $n\sigma^2\ge \(\frac x\sigma\)^{2/k}\ge
M(L+\beta)^{3/2}\log\frac2\sigma$ with $\beta=\max\(\frac{\log D}
{\log n},0\)$.}
\medskip\noindent
{\it Remark:}\/ The introduction of the number $\bar A\ge2^k$ in
Proposition~1 may seem a bit artificial. Its role is to guarantee
that such a number $\bar\sigma$ could be defined in Proposition~1
which satisfies the inequality $\(\frac x{\bar\sigma}\)^{2/k}\ge A
n\bar\sigma^2$ with a sufficiently large previously fixed constant
$A=A(k)$.
\medskip\noindent
{\it Proof of Proposition 1.} For all $p=0,1,2,\dots$ choose a set
$\Cal F_p=\{f_{p,1},\dots,f_{p,m_p}\}\subset\Cal F$ with $m_p\le D\,
2^{2pL}\sigma^{-L}$ elements in such a way that $\inff_{1\le j\le m_p}
\int (f-f_{p,j})^2\,d\mu\le 2^{-4p}\sigma^2$ for all $f\in\Cal F$.
For all pairs $(j,p)$, $p=1,2,\dots$, $1\le j\le m_p$, choose a
predecessor $(j',p-1)$, $j'=j'(j,p)$, $1\le j'\le m_{p-1}$, in such
a way that the functions $f_{j,p}$ and $f_{j',p-1}$ satisfy the
relation $\int|f_{j,p}-f_{j',p-1}|^2\,d\mu\le \sigma^22^{-4(p-1)}$.
Then we have $\int\(\frac{f_{j,p}-f_{j',p-1}}2\)^2\,d\mu\le4
\sigma^2 2^{-4p}$ and $\supp_{x_j\in X,\,1\le j\le k}\left|
\frac{f_{j,p}(x_1,\dots,x_k)-f_{j',p-1}(x_1,\dots,x_k)}2\right|\le 1$.
Inequality (2.1) yields that
$$
\align
P(A(j,p))&=P\(n^{-k/2}|I_{n,k}(f_{j,p}-f_{j',p-1})|\ge
\frac{2^{-(1+p)}x}
{\bar A}\) \le C \exp\left\{-\alpha\(\frac{2^px}{8\bar A
\sigma}\)^{2/k} \right\}\\
&\qquad \text {if}\quad n\sigma^2 2^{-4p}\ge \(\frac {2^px}
{8\bar A\sigma}\)^{2/k},\quad 1\le j\le m_p,\; p=1,2,\dots,\tag2.3
\endalign
$$
and
$$
\aligned
P(B(s))&=P\(n^{-k/2}|I_{n,k}(f_{0,s})|\ge \frac x{2\bar A}\)\le
C\exp\left\{-\alpha\(\frac x{2\bar A\sigma}\)^{2/k}\right\},
\quad 1\le s\le m,\\
&\qquad\qquad\qquad\text{if} \quad n\sigma^2\ge \(\frac x{2\bar
A\sigma}\)^{2/k}.
\endaligned\tag2.4
$$
Choose an integer $R$, $R\ge0$, in such a way that
$2^{(4+{2/k})(R+1)}\(\frac{x}{\bar A\sigma}\)^{2/k} \ge
2^{2+6/k} n\sigma^2 \ge 2^{(4+2/k)R}\(\frac{x}{\bar
A\sigma}\)^{2/k}$, and define $\bar\sigma^2=2^{-4R}\sigma^2$ and
$\Cal F_{\bar\sigma}=\Cal F_R$. (As $n\sigma^2\ge\(\frac
x\sigma\)^{2/k}$ and $\bar A\ge2^k$ by our conditions, there
exists such a non-negative number $R$.) Then the cardinality~$m$
of the set $\Cal F_{\bar\sigma}$ is clearly not greater than
$D\bar\sigma^{-L}$, and $\bigcupp_{j=1}^m \Cal D_j=\Cal F$. Beside
this, the number $R$ was chosen in such a way that inequalities
(2.2) and (2.3) can be applied for $1\le p\le R$. Hence the
definition of the predecessor of a pair $(j,p)$ implies that
$$ \allowdisplaybreaks
\align
&P\(\sup_{f\in\Cal F_{\bar\sigma}}n^{-k/2}|I_{n,k}(f)|
\ge \frac x{\bar A}\) \le P\(\bigcup_{p=1}^R
\bigcup_{j=1}^{m_p}A(j,p) \cup\bigcup_{s=1}^mB(s)\) \\
&\qquad \le \sum_{p=1}^R\sum_{j=1}^{m_p}P(A(j,p))
+\sum_{s=1}^mP(B(s)) \le \sum_{p=1}^{\infty} CD\,2^{2pL}
\sigma^{-L} \exp\left\{-\alpha\(\frac{2^px}
{8\bar A\sigma}\)^{2/k} \right\}\\
&\qquad\qquad +CD\sigma^{-L}\exp\left\{-\alpha\(\frac
x{2\bar A\sigma}\)^{2/k}\right\}.
\endalign
$$
If the condition $\(\frac x\sigma\)^{2/k}\ge ML^{3/2}
\log\frac2\sigma$ holds with a sufficiently large constant
$M$ (depending on $\bar A$), then the inequalities
$$
2^{2pL}\sigma^{-L}\exp\left\{-\alpha\(\frac{2^px}{8\bar
A\sigma}\)^{2/k} \right\}\le 2^{-p} \exp\left\{-\alpha\(\frac{2^{p}x}
{10\bar A \sigma}\)^{2/k} \right\}
$$
hold for all $p=1,2,\dots$, and
$$
\sigma^{-L}\exp\left\{-\alpha\(\frac x{2\bar A\sigma}\)^{2/k}\right\}
\le\exp\left\{-\alpha\(\frac x{10\bar A\sigma}\)^{2/k}\right\}.
$$
Hence the previous estimate implies that
$$
\align
&P\(\sup_{f\in\Cal F_{\bar\sigma}}n^{-k/2}|I_{n,k}(f)|\ge
\frac x{\bar A}\) \le\sum_{p=1}^{\infty}CD 2^{-p}
\exp\left\{-\alpha\(\frac{2^{p}x}{10\bar A \sigma}\)^{2/k}
\right\}\\
&\qquad +CD\exp\left\{-\alpha\(\frac x{10\bar A
\sigma}\)^{2/k}\right\} \le 2CD \exp\left\{-\alpha\(
\frac x{10 \bar A\sigma}\)^{2/k}\right\},
\endalign
$$
and relation (2.2) holds. We have
$$
\align
n\bar\sigma^2&=2^{-4R} n\sigma^2\le
2^{-4R}\cdot2^{(4+2/k)(R+1)-2-6/k}\(\frac{x}{\bar A\sigma}\)^{2/k}=
2^{2-4/k}\cdot 2^{2R/k}\(\frac{x}{\bar A \sigma}\)^{2/k}\\
&=2^{2-4/k}\cdot \(\frac\sigma{\bar\sigma}\)^{1/k}\(\frac{x}{\bar A
\sigma}\)^{2/k}=2^{2-4/k}\cdot \(\frac{\bar\sigma}\sigma\)^{1/k}
\(\frac{x}{\bar A \bar\sigma}\)^{2/k},
\endalign
$$
hence $n\bar\sigma^2\le  4 \(\frac{x}{\bar A\bar\sigma}\)^{2/k}$.
Beside this, as $n\sigma^2\ge 2^{(4+2/k)R-2-6/k}\(\frac u{\bar
A\sigma}\)^{2/k}$, $R\ge1$,
$$
n\bar\sigma^2=2^{-4R}n\sigma^2\ge
2^{-2-6/k}\cdot 2^{2R/k}\(\frac x{\bar
A\sigma}\)^{2/k}\ge\frac1{64}\(\frac x{\bar A\sigma}\)^{2/k}.
$$
It remained to show that $n\bar\sigma^2\ge
\frac{M^{2/3}(L+\beta)\log n}{1000A^{4/3}}$.
 
This inequality clearly holds under the conditions of Proposition~1
if $\sigma\le n^{-1/3}$, since in this case $\log\frac2\sigma\ge
\frac{\log n}3$, and $n\bar\sigma^2\ge\frac1{64}\(\frac x {\bar
A\sigma}\)^{2/k} \ge\frac1{64}\bar A^{-2/k} M(L+\beta)^{3/2}\log
\frac2\sigma\ge \frac1{192}\bar A^{-2/k} M(L+\beta)\log n\ge
\frac{M^{2/3}(L+\beta)\log n}{1000A^{4/3}}$ if $M=M(\bar A,k)$ is
chosen sufficiently large.
 
If $\sigma\ge n^{-1/3}$, then the inequality
$2^{(4+2/k)R}\(\frac x{\bar
A\sigma}\)^{2/k} \le 2^{2+6/k}n\sigma^2$ holds. Hence
$2^{-4R}\ge 2^{-4(2+6/k))/(4+2/k)}  \[\dfrac{\(\frac
x{\bar A\sigma}\)^{2/k}}{n\sigma^2}\]^{4/(4+2/k)}$, and
$$
n\bar\sigma^2=2^{-4R}n\sigma^2\ge \frac{2^{-16/3}}{\bar A^{4/3}}
(n\sigma^2)^{1-\gamma}\[\(\frac x\sigma\)^{2/k}\]^\gamma
\text{ with } \gamma=\frac4{4+\frac2k}\ge\frac23.
$$
Since $n\sigma^2\ge(\frac x\sigma)^{2/k}\ge\frac M3(L+\beta)^{3/2}$,
and $n\sigma^2\ge n^{1/3}$, the above estimates yield that
$n\bar\sigma^2\ge\frac{\bar A^{-4/3}}{50} (n\sigma^2)^{1/3}\[\(\frac
x\sigma\)^{2/k}\]^{2/3}\ge\frac{\bar A^{-4/3}}{50}n^{1/9}\(\frac
M3\)^{2/3} (L+\beta) \ge\frac{M^{2/3}(L+\beta)\log n}{1000 \bar
A^{4/3}}$. \medskip
Now I formulate Proposition~2 and show that Theorem~2 follows from
Propositions~1 and~2.
\medskip\noindent
{\bf Proposition 2.} {\it Let us have a probability measure $\mu$ on
a space $(X,\Cal X)$ together with a sequence of independent and $\mu$
distributed random variables $\xi_1,\dots,\xi_n$ and an $L_2$-dense
class $\Cal F$ of canonical kernel functions $f=f(x_1,\dots,x_k)$
(with respect to the measure~$\mu$) with some parameter $D>0$ and
exponent $L\ge1$ on the product space $(X^k,\Cal X^k)$ which consists
of at most countably many functions, and satisfies conditions (1.3)
and (1.4) with some $0<\sigma\le1$. Let $n\sigma^2>K(L+\beta)\log n$
with $\beta=\max\(\frac{\log D}{\log n},0\)$ and a sufficiently
large constant $K=K(k)$. Then there exist some numbers
$\bar C= \bar C(k)>0$, $\gamma=\gamma(k)>0$ and threshold index
$A_0=A_0(k)>0$ depending only on the order $k$ of the $U$-statistics
we consider such that the degenerate $U$-statistics $I_{n,k}(f)$,
$f\in\Cal F$, defined in (1.6) satisfy the inequality
$$
P\(\sup_{f\in\Cal F}|n^{-k/2}I_{n,k}(f)|\ge A n^{k/2}\sigma^{k+1}\)
\le \bar C e^{-\gamma A^{1/2k}n\sigma^2}\quad \text{if } A\ge A_0.
\tag2.5
$$
} \medskip
 
In the proof of Theorem~2 with the help of Propositions~1
and~2 we exploit our freedom in the choice of the parameters in
these results. Let us choose a number $\bar A_0$ such that
$\bar A_0\ge A_0$ and $\gamma\bar A_0^{1/2k}\ge\frac1K$ with the
numbers $A_0$, $K$ and $\gamma$ in Proposition~2. We shall apply
Proposition~1 with the choice $\bar A=\max (2^{k+2}\bar A_0,2^k)$.
Then by Proposition~1 and the choice of the numbers $\bar A$ and
$\bar A_0$ also the inequality $\(\frac x{\bar\sigma}\)^{2/k}
\ge\frac{\bar A^{2/k}}4n \bar\sigma^2\ge(4\bar A_0)^{2/k}n
\bar\sigma^2$ holds, hence $x\ge 4\bar A n^{k/2}\bar\sigma^{k+1}$
with the number $\bar\sigma$ in Proposition~1. This implies that
$\(\frac12-\frac1{2\bar A}\)x\ge\frac x4\ge\bar An^{k/2}
\bar\sigma^{k+1}$, and $\bar A\ge A_0$. The numbers $x$
considered in these estimations satisfy the condition
$n\sigma^{2/k}\ge \(\frac x\sigma\)^{2/k} \ge
M(L+\beta)^{3/2}\log\frac2\sigma$ imposed in Proposition~1 with
some appropriately chosen constant $M$. Choose the number
$M\ge M(\bar A,k)$ in Proposition~1 (which also can be chosen as
the number~$M$ in formula~(1.7) of Theorem~2) in such a way that
it also satisfies the inequality $\frac{M^{2/3}(L+\beta)\log n}
{1000\bar A^{4/3}}\ge K(L+\beta)\log n$ with the number $K$
appearing in the conditions of Proposition~2. With such a choice
the inequality $n\bar\sigma^2\ge\frac{M^{2/3}(L+\beta)\log n}
{1000\bar A^{4/3}} \ge K(L+\beta)\log n$ holds, and Proposition~2
can be applied with the choice $\bar\sigma$ defined in
Proposition~1 for the parameter $\sigma$, the number
$\(\frac12-\frac1{2\bar A}\)x$ as the number $A$ in this result,
together with the classes of functions $\bar {\Cal D}_j=
\left\{g= \frac{f-f_j}2,\,f\in\Cal D_j\right\}$, $1\le j\le m$,
where the classes of functions $\Cal D_j$ and functions $f_j$,
$1\le j\le m$, are defined in Proposition~1.
 
Then Propositions~1 and~2 together with the above observations
yield that
$$ \allowdisplaybreaks
\align
&P\(\supp_{f\in\Cal F}n^{-k/2}|I_{n,k}(f)|\ge x\)\le
P\(\sup_{f\in\Cal F_{\bar\sigma}}n^{-k/2}|I_{n,k}(f)|\ge
\frac x{\bar A}\) \\
&\qquad\qquad +\sum_{j=1}^m P\(\sup_{g\in\Cal D_j} n^{-k/2}
\left|I_{n,k}\(\frac{f_j-g}2\)\right| \ge\(\frac12-\frac1{2\bar
A}\)x\)  \tag2.6   \\
&\le 2CD\exp\left\{-\alpha\(\frac x{10\bar A\sigma}\)^{2/k}\right\}
+\bar CD\bar\sigma^{-L} e^{-\gamma\bar A^{1/2k}n\bar\sigma^2}.
\endalign
$$
To get the result of Theorem~2 from inequality (2.6) we have to
replace its second term at the right-hand side with a more
appropriate expression where, in particular, we get rid of the
coefficient $\bar\sigma^{-L}$. The condition
$n\bar\sigma^2\ge K(L+\beta)\log n$ implies that $\bar\sigma\ge
n^{-1/2}$, and by our choice of $\bar A_0$ we have $\gamma \bar
A_0^{1/2k}n\bar\sigma^2\ge \frac1Kn\bar\sigma^2 \ge L\log n\ge
2L\log\frac1{\bar \sigma}$, i.e. $\bar\sigma^{-L}\le e^{\gamma\bar
A_0^{1/2k}n\bar\sigma^2/2}$. By the estimates of Proposition~1
$n\bar\sigma^2 \ge\frac1{64}\(\frac x{\bar A\sigma}\)^{2/k}$. The
above relations imply that $\bar\sigma^{-L} e^{-\gamma\bar
A_0^{1/2k}n \bar\sigma^2}\le e^{-\gamma\bar
A_0^{1/2k}n\bar\sigma^2/2}\le
\exp\left\{-\frac\gamma{128} \bar A_0^{1/2k} \bar A^{-2/k}\(\frac
u\sigma\)^{2/k}\right\}$. Hence relation (2.6) yields that
$$
\align
&P\(\supp_{f\in\Cal F}n^{-k/2}|I_{n,k}(f)|\ge x\)\\
&\qquad\le 2CD\exp \left\{-\frac\alpha{(10\bar A)^2}\(\frac
x\sigma\)^{2/k}\right\} +\bar CD\exp\left\{-\frac\gamma{128}
\bar A_0^{1/2k} \bar A^{-2/k} \(\frac x\sigma\)^{2/k}\right\},
\endalign
$$
and this estimate implies Theorem~2 with some new appropriately
defined constants $\alpha>0$ and $C>0$.
\medskip
 
Thus I have reduced the proof of Theorem~2 to that of
Proposition~2. I also show in this section that the proof of
Theorem~1 can be reduced to that of Theorem~2 and a decomposition
result of random integrals $J_{n,k}(f)$ formulated in Proposition~3
below whose proof will be given in the Appendix. Proposition~3
gives the representation of a random integral $J_{n,k}(f)$ in the
form of a linear combination of degenerate $U$-statistics. To
get this representation we can observe that a random integral
$J_{n,k}(f)$ can be rewritten in the form of a sum of
$U$-statistics. By applying an important result, called Hoeffding's
decomposition, we can write a general $U$-statistic in the form of
a sum of degenerate $U$-statistics of different order. Proposition~3
contains the result we get by carrying out this procedure. Let us
recall that we have integrated with respect to the signed measure
$\mu_n-\mu$ in the definition~(1.3) of the random integrals
$J_{n,k}(f)$. This has a very strong cancellation effect, and the
main content of Proposition~3 is that this implies that the
representation of $J_{n,k}(f)$ in the form of a linear combination
of degenerate $U$-statistics contains small coefficients.
 
Beside Proposition~3 we need another result to deduce Theorem~1
from Theorem~2. We must have some control on the exponent and parameter
of the classes of functions appearing in the Hoeffding decomposition
of the class of functions we consider together with a good
$L_2$-norm of these functions. Hoeffding's
decomposition is made with the help of certain projections introduced
in formulas~(2.7) and~(2.8) below. In Lemma~1 I prove the
properties of these projections I shall need later. I shall need
Lemma~1 also in the proof of Proposition~2, since
Hoeffding's decomposition is applied in it.
 
Let some measurable spaces $(Y_1,\Cal Y_1)$, $(Y_2,\Cal Y_2)$ and
$(Z,\Cal Z)$ be given together with a probability measure $\mu$
on the space $(Z,\Cal Z)$. Consider a function $f(y_1,z,y_2)$ on
the product space $(Y_1\times Z\times Y_2,\Cal Y_1\times\Cal Z
\times \Cal Y_2)$, $y_1\in\Cal Y_1$, $z\in\Cal Z$, $y_2\in\Cal Y_2$,
and define their projections
$$
P_\mu f(y_1,y_2)=\int f(y_1,z,y_2)\mu(\,dz),\quad y_1\in Y_1,\;
y_2\in Y_2, \tag2.7
$$
and
$$
\aligned
Q_\mu f(y_1,z,y_2)&=(I-P_\mu) f(y_1,z,y_2)\\
&=f(y_1,z,y_2)-P_\mu f(y_1,z,y_2), \quad
y_1\in Y_1,\;z\in Z,\;y_2\in Y_2,
\endaligned \tag2.8
$$
where $P_\mu f(y_1,z,y_2)=P_\mu f(y_1,y_2)$, i.e. I have introduced
a fictive argument $z$ of the function $\bar P_\mu f$ in formula~(2.8)
to make it meaningful. Now I formulate the following
\medskip\noindent
{\bf Lemma~1.} {\it Let us have some measurable spaces $(Y_1,\Cal Y_1)$,
$(Y_2,\Cal Y_2)$ and $(Z,\Cal Z)$, a probability measure $\mu$ on
the space $(Z,\Cal Z)$ and a probability measure $\rho$ on the product
space $(Y_1\times Y_2,\Cal Y_1\times\Cal Y_2)$. The transformations
$P_\mu$ and $Q_\mu$ defined in (2.7) and (2.8) are
contractions from the space $L_2(Y_1\times Z\times Y_2,\rho\times\mu)$
to the spaces $L_2(Y_1\times Y_2,\rho)$ and
$L_2(Y_1\times Z\times Y_2,\rho\times\mu)$ respectively, i.e.
$$
\aligned
\|P_\mu f\|_{L_2,\rho}^2&=\int P_\mu
f(y_1,z,y_2)^2\rho(\,dy_1,\,dy_2) \\
&\le\|f\|_{L_2,\rho\times \mu}^2= \int
f(y_1,z,y_2)^2\rho(\,dy_1,\,dy_2)\mu(\,dz),
\endaligned \tag2.9
$$
and
$$
\aligned
\|Q_\mu f\|_{L_2,\rho}^2&=\int Q_\mu
f(y_1,z,y_2)^2\rho(\,dy_1,\,dy_2) \\
&=\int\(f(y_1,z,y_2)-P_\mu
f(y_1,z,y_2)\)^2\rho(\,dy_1,\,dy_2)\mu(\,dz)\\
&\le\|f\|_{L_2,\rho\times \mu}^2= \int
f(y_1,z,y_2)^2\rho(\,dy_1,\,dy_2)\mu(\,dz).
\endaligned \tag$2.9'$
$$
Also the inequalities
$$
\align
\sup_{y_1,y_2} |P_\mu f(y_1,y_2)|&\le
\sup_{y_1,z,y_2}|f(y_1,z,y_2)| \tag2.10 \\
\sup_{y_1,z,y_2} |Q_\mu f(y_1,z,y_2)|&\le2\sup_{y_1,z,y_2}
|f(y_1,z,y_2)| \tag$2.10'$
\endalign
$$
hold. If $\Cal F$ is an $L_2$-dense class of functions
$f(y_1,z,y_2)$ on the product space $(Y_1\times Z\times Y_2,
\Cal Y_1\times\Cal Z\times Y_2)$, $y_1\in\Cal Y_1$, $z\in\Cal Z$,
$y_2\in\Cal Y_2$ with parameter $D$ and exponent $L$, then also the
classes $\Cal F_\mu=\{P_\mu f,\colon f\in \Cal F\}$ with the functions
$P_\mu f$ defined in formulas (2.7) are $L_2$-dense classes with
parameter~$D$ and exponent~$L$ in the space $(Y_1\times Y_2,
\Cal Y_1\times\Cal Y_2)$. Beside this, the class of functions
$\Cal G_\mu=\{\frac12Q_\mu f=\frac12(f-P_\mu f),\; f\in\Cal F\}$
is also an $L_2$-dense class with exponent $L$ and parameter~$D$.}
\medskip\noindent
{\it Proof of Lemma 1.}\/ The Schwarz inequality yields that
$P_\mu(f)^2\le\int f(y_1,z,y_2)^2\mu(\,dz)$, and the inequality
$\int [f(y_1,z,y_2)-P_\mu f(y_1,z,y_2)]^2\mu(dz)\le \int
f(y_1,z,y_2)^2\mu(\,dz)$ also holds. Integrating these inequalities
with respect to the probability measure $\rho(\,dy_1,\,dy_2)$ we
get formulas~(2.9) and~($2.9'$). The proof of relations~(2.10)
and~$(2.10')$ is self-evident.
 
Let us consider an arbitrary probability measure $\rho$ on the space
$(Y_1\times Y_2,\Cal Y_1\times\Cal Y_2)$. To prove that $\Cal F_\mu$
is an $L_2$-dense class with exponent~$L$ and parameter~$D$ we have
to find $m\le D \e^L$ functions $f_j\in \Cal F_\mu$, $1\le j\le m$,
such that $\inff_{1\le j\le m}\int (f_j-f)^2\,d\rho\le \e^2$ for all
$f\in \Cal F_\mu$. We can find such a sequence, since a similar
statement holds for the class of functions $\Cal F$ in the space
$Y_1\times Z\times Y_2$ with the probability measure
$\rho\times\mu$. This fact together with the $L_2$ contraction
property of $P_\mu$ formulated in (2.9) imply that $\Cal F_\mu$ is
an $L_2$-dense class.
 
The $L_2$-density property of the set $\Cal G_\mu$ under the
appropriate conditions can be deduced from the following
observation. For any probability measure $\rho$ on the space
$Y_1\times Z\times Y_2$ and pair of functions $f$ and $g$ such
that $\int(f-g)^2\frac12\(\,d\rho+\,d\bar\rho\times\,du\)\le \e^2$,
where $\bar\rho$ is the projection of the measure $\rho$ to the
space $Y_1\times Y_2$,  i.e. $\bar\rho(A)=\rho(A\times Z)$ for all
$A\in\Cal Y_1\times\Cal Y_2$, the inequality
$\int ((f-P_\mu f)-(g-P_\mu g))^2\,d\rho\le2\int (f-g)^2\,d\rho
+2\int (P_\mu f-P_\mu g)^2\,d\rho\le2\int (f-g)^2\,d\rho+2\int
(f-g)^2\,d\bar\rho\times d\mu\le4\e^2$ holds. This means that if
$\{f_1,\dots,f_m\}$ is an $\e$-dense subset of $\Cal F$ in the
space $L_2(Y_1\times Z\times Y_2,\Cal Y_1\times \Cal Z\times \Cal
Y_2,\tilde\rho)$ with $\tilde\rho=\frac12(\rho+\bar\rho\times\mu)$,
then $\{Q_\mu f_1,\dots,Q_\mu f_m\}$ is a $2\e$-dense subset of
$2\Cal G_\mu=\{f-P_\mu f\colon f\in\Cal F\}$ in the space $L_2(Y_1\times
Z\times Y_2,\Cal Y_1\times \Cal Z\times \Cal Y_2,\rho)$. Hence, if
$\{f_1,\dots,f_m\}$ is an $\e$-dense subset with respect to the
measure $\tilde\rho=\frac12\(\rho+\bar\rho\times\mu\)$, then
$\{\frac12Q_\mu f_1,\dots,\frac12Q_\mu f_m\}$ is an $\e$-dense
subset of $\Cal G_\mu$ in the space $L_2(Y_1\times Z\times
Y_2,\Cal Y_1\times \Cal Z\times \Cal Y_2,\rho)$ space.
\medskip
 
To formulate Proposition 3 first I introduce the following notation.
Given a function $f(x_1,\dots,x_k)$ of $k$ variables on $(X^k,\Cal
X^k)$ together with some probability measure $\mu$ let us introduce
for all sets $V\subset\{1,\dots,k\}$ the function $f_V$ depending on
the arguments $x_j$, $j\in V$ by the formulas
$$
f_V(x_s,\,s\in V)=\(\prod_{s\in\{1,\dots,k\}\setminus V}P_{\mu,s}
\prod_{s\in V}Q_{\mu,s}\)f(x_1,\dots,x_k), \tag2.11
$$
where  $P_{\mu,s}$ and $Q_{\mu,s}$ denote the operators $P_\mu$ and
$Q_\mu$ defined in formulas (2.7) and (2.8) in the space $(Y_1\times
Z\times Y_2,\Cal Y_1\times \Cal Z\times\Cal Y_2)$, where $(Y_1,\Cal
Y_1)$ the product of the first $s-1$, $(Y_2,\Cal Y_2)$ the
product of the last $k-s$ coordinates, and $(Z,\Cal Z)$ is the $s$-th
coordinate of the product space $(X^k,\Cal X^k)$. The function $f_V$
depends only on the coordinates $x_s$, $s\in V$, because at the
application of the operator $P_{\mu,s}$ the $s$-th coordinate
disappears. It can be shown that the function $f_V$ is canonical. To
see this we have to observe that the canonical property of the
function $f_V$ can be reformulated as $P_{\mu,s}f_V\equiv0$ for all
$s\in V$. Beside this, the operator $P_{\mu,s}$ or $Q_{\mu,s}$ is
exchangeable with $P_{\mu,s'}$ or $Q_{\mu,s'}$ if $s\neq s'$, and
$P_{\mu,s}Q_{\mu,s}=P_{\mu,s}-P_{\mu,s}^2=0$.
The functions $f_V$ defined in (2.11) appear
in the Hoeffding decomposition of a $U$-statistic with kernel
function~$f$.
 
Now I formulate Proposition 3 which will be proved in the Appendix.
\medskip\noindent
{\bf Proposition 3.} {\it Let us have a non-atomic measure $\mu$
on a measurable space $(X,\Cal X)$ together with a sequence of
independent, $\mu$-distributed random variables $\xi_1,\dots,\xi_n$,
and take a function $f(x_1,\dots,x_k)$ of $k$ variables on the
space $(X^k,\Cal X^k)$ such that
$$
\int f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)<\infty.
$$
Let us consider the empirical distribution function $\mu_n$ of the
sequence $\xi_1,\dots,\xi_n$ introduced in (1.1) together with the
$k$-fold random integral $J_{n,k}(f)$ of the function $f$ defined in
(1.2). The identity
$$
J_{n,k}(f)=\sum_{V\subset\{1,\dots,k\}}C(n,k,V)n^{-|V|/2}
I_{n,|V|}(f_V), \tag2.12
$$
holds with the canonical (with respect to the measure $\mu$)
functions $f_V(x_j,\;j\in V)$  defined in (2.11) and appropriate
real numbers $C(n,k,V)$, $V\subset\{1,\dots,k\}$, where
$I_{n,|V|}(f_V)$ is the (degenerate) $U$-statistic with kernel
function $f_V$ and random sequence $\xi_1,\dots,\xi_n$ defined in
(1.6). The constants $C(n,k,V)$ in~(2.12) satisfy the relations
$|C(n,k,V)|\le C(k)$ with some constant $C(k)$ depending only on
the order $k$ of the integral $J_{n,k}(f)$,
$\limm_{n\to\infty}C(n,k,V)=C(k,V)$ with some constant
$C(k,V)<\infty$ for all $V\subset\{1,\dots,k\}$, and
$C(n,k,\{1,\dots,k\})=1$ for $V=\{1,\dots,k\}$. }
\medskip
 
Theorem~1 can be simply deduced from Theorems~2, Proposition~3 and
Lemma~1. Indeed, Lemma~1 together with formula~(2.11) imply that if
$\Cal F$ is an $L_2$-dense class of functions with exponent~$L$ and
parameter~$D$, and the elements of $\Cal F$ satisfy relations (1.3)
and~(1.4) with some $\sigma>0$, then for all $V\subset\{1,\dots,k\}$
the class of functions $\Cal F_V=\{2^{-|V|}f_V\colon f\in \Cal F\}$,
where $f_{V}$ is defined in (2.11), and $|V|$ denotes the cardinality of
the set $V$ is again $L_2$-dense with exponent~$L$ and parameter~$D$,
whose elements satisfy relations~(1.3) and~(1.4) with
parameter $2^{-|V|}\sigma$. Beside this, the elements of $\Cal F_V$
are canonical functions. Hence, by Proposition~3 we can write
$$
P\(\sup_{f\in\Cal F}|J_{n,k}(f)|>x\)\le \sum_{V\subset\{1,\dots,k\}}
P\(\sup_{f\in\Cal F}n^{-|V|/2}|I_{n,|V|}(f_V)|>\frac x{2^kC(k)}\)
\tag2.13
$$
with a constant $C(k)$ satisfying the inequality $C(n,k,|V|)\le
C(k)$ for all coefficients $C(n,k,|V|)$ in~(2.12),
and each term at the right-hand side of~(2.13) can be estimated by
means of Theorem~2 if $\Cal F$ satisfies the conditions of Theorem~1.
 
Theorem 1 with appropriate universal constants $M>0$, $C>0$ and
$\alpha>0$ can be proved with the help of some calculation if we
bound each probability on the right-hand side of (2.13)  by means
of Theorem~2. Let me remark that Theorem~1 implicitly contains the
condition that $n\sigma^2\ge M(L+\beta)^{3/2}\log\frac2\sigma$, which
means that the set of numbers $x$ which satisfy the condition in
relation (1.5) is not empty. Hence we may assume that $n\sigma^2\ge1$.
We need this observation to check that under the conditions of
Theorem~1 $n\sigma^2\ge\(\frac x\sigma\)^{2/l}$ for all $l\le k$,
and we can apply Theorem~1 for each term $V\subset\{1,\dots,k\}$ in
the estimation of the right-hand side of~(2.13).
 
\beginsection 3. Some basic tools of the proof of Proposition 2
 
I shall prove Proposition~2 by means of some symmetrization procedure.
The proof becomes simpler with the help of a decoupling
argument. This means the introduction of decoupled $U$-statistics
and the proof of a version of Proposition~2 about decoupled
$U$-statistics. It can be shown with the help of some known
results that this statement implies Proposition~2 in its original
form. To carry out such a program first I recall the definition of
decoupled $U$-statistics.
\medskip\noindent
{\bf Definition of decoupled $U$-statistics.} {\it Let $k$
independent copies $\xi_{1,s}$,\dots,~$\xi_{n,s}$,
$1\le s\le k$, of a sequence of independent and identically
distributed random variables $\xi_1,\dots,\xi_n$ with
distribution~$\mu$ be given on a measurable space $(X,\Cal X)$
together with a function $f=f(x_1,\dots,x_k)$ on the $k$-th power
$(X^k, \Cal X^k)$ of the space $(X,\Cal X)$. We define with their
help the decoupled $U$-statistic~$\bar I_{n,k}(f)$ of order $k$
with kernel function~$f$ by the formula
$$
\bar I_{n,k}(f)=\frac1{k!}\summ\Sb 1\le j_s\le n,\; s=1,\dots, k\\
j_s\neq j_{s'} \text{ if } s\neq s'\endSb
f\(\xi_{j_1,1},\dots,\xi_{j_k,k}\). \tag3.1
$$
A decoupled $U$-statistic is called degenerate if its kernel
function is canonical.}
\medskip
I shall prove the following version of Proposition~2.
 
\medskip\noindent
{\bf Proposition~$2'$.} {\it Let us have a probability measure
$\mu$ on a space $(X,\Cal X)$ together with $k$ independent copies
$\xi_{1,s},\dots,\xi_{n,s}$, $1\le s\le k$, of a sequence of
independent and $\mu$ distributed random variables $\xi_1,\dots,\xi_n$
and a countable $L_2$-dense class $\Cal F$ of canonical kernel
functions $f=f(x_1,\dots,x_k)$ (with respect to the measure~$\mu$)
with some parameter $D>0$ and exponent $L\ge1$ on the product
space $(X^k,\Cal X^k)$ which satisfies conditions (1.3) and (1.4)
with some $0<\sigma\le1$. Let $n\sigma^2>K(L+\beta)\log n$ with
$\beta=\max\(\frac{\log D}{\log n},0\)$ and a sufficiently large
constant $K=K(k)$. There exists some threshold index
$A_0=A_0(k)>0$ such that the decoupled $U$-statistics
$\bar I_{n,k}(f)$, $f\in\Cal F$, defined in (3.6) satisfy the
inequality
$$
P\(\sup_{f\in\Cal F}|n^{-k/2}\bar I_{n,k}(f)|\ge An^{k/2}\sigma^{k+1}\)
\le e^{-A^{1/2k}n\sigma^2}\quad \text{if }A\ge A_0. \tag3.2
$$
} \medskip
 
Proposition 2 follows from Proposition~$2'$ and the following
Proposition A.
\medskip\noindent
{\bf Proposition A.} {\it Let us consider a countable sequence
$f_l(x_1,\dots,x_k)$, $l=1,2,\dots$, of functions on the $k$-fold
product $(X^k,\Cal X^k)$ of some space $(X,\Cal X)$ together with
some probability measure $\mu$ on the space $(X,\Cal X)$. Given a
sequence of independent and identically distributed random variables
$\xi_1,\xi_2,\dots$ with distribution~$\mu$ on $(X,\Cal X)$ together
with $k$ independent copies $\xi_{1,s},\xi_{2,s},\dots$,
$1\le s\le k$, of it we can define the $U$-statistics $I_{n,k}(f_l)$
and decoupled $U$-statistics $\bar I_{n,k}(f_l)$ for all
$l=1,2,\dots$ and $n=1,2,\dots$. They satisfy the inequality
$$
P\(\sup_{1\le l<\infty} \left| I_{n,k}(f_l)\right|>x\)\le
AP\(\sup_{1\le l<\infty}\left|\bar I_{n,k}(f_l)\right|>\gamma x\)
\tag3.3
$$
for all $x\ge0$ with some constants $A=A(k)>0$ and $\gamma
=\gamma(k)>0$ depending only on the order $k$ of the $U$-statistics.}
\medskip
 
Proposition~A can be deduced from Theorem~1 in paper~[6] of de la
Pe\~na and Mont\-go\-mery--Smith which compares the distribution
of a single $U$-statistic with its decoupled $U$-statistic
counterpart. It holds for $U$-statistics with a kernel function
taking values in a general separable Banach space, and it compares
the distribution of the norm of a $U$-statistic with its decoupled
counterpart. This result states that formula~(3.3) remains valid if
we fix a function $f$ of $k$-variables taking values in a separable
Banach space and replace $\sup |I_{n,k}(f_l)|$ by $\|I_{n,k}(f)\|$
and $\sup |\bar I_{n,k}(f_l)|$  by $\|\bar I_{n,k}(f)\|$. Moreover,
the universal constants $A$ and $\gamma$ do not depend on the Banach
space, where the function~$f$ takes its values. In the proof of
Proposition~A we exploit our freedom to work in an arbitrary
separable Banach space.
\medskip\noindent
{\it The proof of Proposition A (with the help of paper~[6].)}\/
Let us fix an arbitrary positive integer $N$, and apply the first
part of Theorem 1 of~[6] in the Banach space $\ell_\infty^N$
consisting of the sequences $x=(x_1,\dots,x_N)$ of length~$N$ of
real numbers with norm $\|x\|=\!\!\supp_{1\le l<N} \!|x_l|$ for
the $U$-statistic and degenerate $U$-statistic with kernel
functions $f_{j_1,\dots,j_k}(x_1,\dots,x_k)=
\bar f(x_1,\dots,x_k)$, $\bar f=(f_1,\dots,f_N)$, with the functions
$f_l$, $1\le l\le N$, in Proposition~A, which maps the space
$(X^k,\Cal X^k)$ into the space $\ell_\infty^N$.
(Here we do not exploit that in the result of~[6] the kernel
functions may depend on the indices $(j_1,\dots,j_k)$.)
The first part of Theorem~1 in~[6] states that
$$
\aligned
&P\(\left\|\summ\Sb 1\le j_s\le n,\; s=1,\dots, k\\
j_s\neq j_{s'} \text{ if } s\neq s'\endSb
\bar f\(\xi_{j_1},\dots,\xi_{j_k}\)\right\|>x\) \\
&\qquad \le AP\(\left\| \summ\Sb 1\le j_s\le n,\; s=1,\dots, k\\
j_s\neq j_{s'} \text{ if } s\neq s'\endSb
\bar f\(\xi_{j_1,1},\dots,\xi_{j_k,k}\)\right\|>\gamma x\)
\endaligned  \tag3.4
$$
with some universal constants $A=A(k)>0$ and $\gamma=\gamma(k)>0$,
and this statement is equivalent to a weaker version of
relation~(3.3), where $\supp_{1\le l<\infty}$ is replaced by
$\supp_{1\le l\le N}$. We get relation (3.3) from relation~(3.4) by
letting $N\to\infty$ (and exploiting that the constants $A$ and
$\gamma$ in formula~(3.4) do not depend on the number $N$.)
\medskip\noindent
{\it Remark:}\/ I have introduced the number $N$ in the above proof
instead of working in the space of infinite sequences with $L_\infty$
norm to avoid the difficulty which would arise if we had to work in
non-separable Banach spaces.
\medskip
 
Thus I have reduced the proof of Theorem~2 to that of
Proposition~$2'$. It will be proved by means of a symmetrization
argument. To apply this argument I shall need two auxiliary results,
the multi-dimensional version of Hoeffding's inequality and an
appropriate generalization of a well-known symmetrization lemma.
 First I discuss the multi-dimensional version
of Hoeffding's inequality.
 
\medskip\noindent
{\bf Lemma 2. (The multi-dimensional version of Hoeffding's
inequality.)} {\it Let $\e_1,\dots,\e_n$ be independent
random variables, $P(\e_j=1)=P(\e_j=-1)=\frac12$, $1\le j\le n$.
Fix a positive integer~$k$, and define the random variable
$$
Z=\sum\Sb (j_1,\dots, j_k)\colon 1\le j_l\le n \text{ for all } 1\le
l\le k\\ j_l\neq j_{l'} \text{ if }l\neq l' \endSb a(j_1,\dots, j_k)
\e_{j_1}\cdots \e_{j_k} \tag3.5
$$
with the help of some real numbers $a(j_1,\dots,j_k)$ which are given
for all sets of indices such that $1\le j_l\le n$, $1\le l\le k$, and
$j_l\neq j_{l'}$ if $l\neq l'$. Put
$$
S^2=\sum\Sb (j_1,\dots, j_k)\colon 1\le j_l\le n \text{ for all } 1\le
l\le k\\ j_l\neq j_{l'} \text{ if }l\neq l' \endSb a^2(j_1,\dots, j_k).
\tag3.6
$$
Then
$$
P(|Z|>x)\le C \exp\left\{-B\(\frac xS\)^{2/k}\right\} \quad\text{for
all }x\ge 0 \tag3.7
$$
with some constants $B>0$ and $C>0$ depending only on the parameter
$k$. Relation (3.5) holds for instance with the choice
$B=\frac k{2e(k!)^{1/k}}$ and $C=e^k$.}
\medskip
Lemma~2 is a relatively simple consequence of an important result
of the probability theory, the hypercontractive inequality for
Rademacher functions~(see~e.g.~[4] or~[5]). It yields some moment
inequalities that imply Lemma~2. Such an inequality is formulated
e.g. in Theorem~3.2.2 of~[5]. It states (with the choice $p=2$ in
this result and the observation $EZ^2\le k!S^2$) that
$$
E|Z|^q\le (q-1)^{kq/2}(k!S^2)^{q/2} \qquad \text{for } q\ge2.
\tag3.8
$$
Here I used the notation of Lemma 2.
 
The Markov inequality and inequality (3.8) imply that
$$
P(|Z|>x)\le \(q^{k/2}\frac {\sqrt{k!}S}x\)^q   \quad \text{for all
}x>0\quad \text{and } q\ge2.
$$
Choose the number $q$ as the solution of the equation
$q\(\frac {\sqrt{k!}S}x\)^{2/k}=\frac1e$. Then we get that
$P(|Z|>x)\le \exp\left\{- B\(\frac xS\)^{2/k}\right\}$ with
$B=\frac k{2e(k!)^{1/k}}$, provided that $q=\frac1{e{k!}^{1/k}}
\(\frac xS\)^{2/k}\ge2$, i.e. $B\(\frac xS\)^{2/k}\ge k$. By
multiplying the above upper bound with $C=e^k$ we get such an
estimate for $P(|Z|>x)$ which holds for all $x>0$. In such a way
we get the proof of Lemma~2.
 
\medskip\noindent
{\it Remark:}\/ The parameter $B$ given in Lemma~2 is not
sharp. In paper~[9] I have shown that the right choice of $B$
in formula (3.7) is $B=\frac12$.
\medskip\noindent
 
The second result I need is a slight generalization of a simple
lemma that can be found for instance in Pollard's book~[12]
($8^\circ$ Symmetrization Lemma) or Lemma~2.5 in~[7]. In this paper
I need the result given in Lemma~3 below to carry out my arguments.
Its proof consists of a slight modification of the method in~[7]
or~[12].
\medskip\noindent
{\bf Lemma 3. (Symmetrization Lemma)} {\it Let $Z(n)$ and $\bar Z(n)$,
$n=1,2,\dots$, be two sequences of random variables on a probability
space $(\Omega,\Cal A,P)$. Let a $\sigma$-algebra $\Cal B\subset
\Cal A$ be given on the probability space $(\Omega,\Cal A,P)$ together
with a $\Cal B$ measurable set $B$ and two numbers $\alpha>0$ and
$\beta>0$ such that the random variables $Z_n$, $n=1,2,\dots$, are
$\Cal B$ measurable, and the inequality
$$
P(|\bar Z_n|\le\alpha|\Cal B)(\oo)\ge\beta\quad \text{for all }
n=1,2,\dots \text{ if } \oo\in B \tag3.9
$$
holds.
Then
$$
P\(\sup_{1\le n<\infty}|Z_n|>\alpha+x\)\le\frac1\beta P\(\supp_{1\le
n<\infty}|Z_n-\bar Z_n|>x\)+(1-P(B))\quad\text{for all } x>0.
\tag3.10
$$
In particular, if the sequences $Z_n$, $n=1,2,\dots$, and $\bar Z_n$,
$n=1,2,\dots$, are two independent sequences of random  variables,
and $P(|\bar Z_n|\le\alpha)\ge\beta$ for all $n=1,2,\dots$, then
$$
P\(\sup_{1\le n<\infty}|Z_n|>\alpha+x\)\le\frac1\beta P\(\supp_{1\le
n<\infty}|Z_n-\bar Z_n|>x\). \tag$3.10'$
$$
}\medskip\noindent
{\it Proof of Lemma 3.}\/ Put $\tau=\min\{n\colon |Z_n|>\alpha+x\}$ if
there exists such an $n$, and $\tau=0$ otherwise. Then
$$
\align
P(\{\tau=n\}\cap B)&\le \frac1\beta\int_{\{\tau=n\}\cap B} P(|\bar
Z_n|\le \alpha|\Cal B)\,dP
=\frac1\beta P(\{\tau=n\}\cap\{|\bar Z_n|\le\alpha\}\cap B)\\
&\le \frac1\beta P(\{\tau=n\}\cap\{|Z_n-\bar Z_n|>x\})
\quad \text{for all } n=1,2,\dots.
\endalign
$$
Hence
$$
\align
&P\(\sup_{1\le n<\infty}|Z_n|>\alpha+x\)-(1-P(B))\le
P\(\left\{\sup_{1\le n<\infty}|Z_n|>\alpha+x\right\}\cap B\) \\
&\qquad=\sum_{n=1}^\infty P(\{\tau=n\}\cap B)
\le \frac1\beta \sum_{n=1}^\infty P(\{\tau=n\}\cap\{|Z_n-\bar
Z_n|>x\}) \\
&\qquad \le\frac1\beta P\(\supp_{1\le n<\infty}|Z_n-\bar Z_n|>x\).
\endalign
$$
Thus formula 3.10 is proved. If $Z_n$ and $\bar Z_n$ are two
independent sequences, and $P(|\bar Z_n|\le\alpha)\ge\beta$ for all
$n=1,2,\dots$, and we define $\Cal B$ as the $\sigma$-algebra
generated by the random variables $Z_n$, $n=1,2,\dots$, then
condition (3.9) is satisfied also with $B=\Omega$. Hence relation
($3.10'$) holds in this case.  Lemma~3 is proved.
\medskip
 
Before turning to the proof of Proposition~$2'$ I explain the main
ideas of its proof. These ideas are taken from the paper~[1] of
Alexander.
 
Let us restrict our attention to the case~$k=1$. In this case a
probability of the form $P\(n^{-1/2}\supp_{f\in\Cal F}
\left|\summ_{j=1}^n f(\xi_j)\right|>x\)$ has to be estimated. By
taking an independent copy of the sequence $\xi_n$ (which disappears
at the end of the of the calculation) a symmetrization argument can
be applied which reduces the problem to the estimation of the
probability $P\(n^{-1/2}\supp_{f\in\Cal F}\left|
\summ_{j=1}^n \e_jf(\xi_j)\right|>\bar x\)$, where the random
variables $\e_j$, $P(\e_j=1)=P(\e_j=-1)=\frac12$, $j=1,\dots,n$, are
independent, and they are independent also of the random variables
$\xi_j$. Beside this, the number $\bar x$ is only slightly smaller
than the number~$x/2$. Let us bound the conditional probability of
the event we have just introduced if the values random variables
$\xi_j$ are prescribed in it. This conditional probability can be
bounded by means of the one-dimensional version of Lemma~2,
and the estimate we get in such a way is useful if the conditional
variance of the random variable we have to handle has a good upper
bound. Such a bound exists, and some calculation reduces the original
problem to the estimation of the probability
$P\(n^{-1/2}\supp_{f\in\Cal F'}\left| \summ_{j=1}^n
f(\xi_j)\right|>x^{1+\alpha}\)$ with some new nice class of functions
$\Cal F'$ and number $\alpha>0$. This problem is very similar to the
original one, but it is simpler, since the number $x$ is replaced by
a larger number $x^{1+\alpha}$ in it. By repeating this argument
successively, in finitely many steps we get to an
inequality that clearly holds.
 
The above sketched argument suggests a backward induction procedure
to prove Proposition~$2'$. To carry out such a program I shall prove
a result formulated in Proposition~4. First I introduce
the following notion.
\medskip\noindent
{\bf Definition of good tail behaviour for a class of $U$-statistics.}
{\it Let us have some measurable space $(X,\Cal X)$ and a probability
measure $\mu$ on it. Let us consider some class $\Cal F$ of functions
$f(x_1,\dots,x_k)$ on the $k$-fold product $(X^k,\Cal X^k)$ of the
space $(X,\Cal X)$. Fix some positive integer~$n$ and
positive number $\sigma>0$, and take $k$ independent
copies $\xi_{1,s},\dots,\xi_{n,s}$, $1\le s\le k$, of a
sequence of independent $\mu$-distributed random variables
$\xi_1,\dots,\xi_n$. Let us introduce with the help of these random
variables the decoupled $U$-statistics $\bar I_{n,k}(f)$, $f\in\Cal
F$. Given some real number $T>0$ we say that the set of decoupled
$U$-statistics determined by the class of functions $\Cal F$ has a
good tail behaviour at level~$T$ if the following inequality holds:
$$
P\(\sup_{f\in\Cal F}|n^{-k/2}\bar I_{n,k}(f)|\ge A
n^{k/2}\sigma^{k+1}\) \le \exp\left\{-A^{1/2k}n\sigma^2 \right\}
\quad \text{for all } A\ge T.  \tag3.11
$$
} \medskip
Now I formulate Proposition 4 which enables us to make the
inductive procedure leading to the proof of Proposition~$2'$.
\medskip\noindent
{\bf Proposition 4.} {\it Let us fix a positive integer~$n$, real
number $0<\sigma\le2^{-(k+1)}$ and a probability measure $\mu$ on
a measurable space $(X,\Cal X)$ together with a countable $L_2$-dense
class $\Cal F$ of canonical kernel functions $f=f(x_1,\dots,x_k)$
(with respect to the measure~$\mu$) on the $k$-fold product space
$(X^k,\Cal X^k)$ with some exponent $L\ge1$ and parameter~$D>0$.
Let us also assume that all functions $f\in \Cal F$ satisfy the
conditions $\supp_{x_j\in X, 1\le j\le k}|f(x_1,\dots,x_k)|\le
2^{-(k+1)}$, $\int f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)
\le\sigma^2$, and $n\sigma^2>K(L+\beta)\log n$ with a sufficiently
large fixed number $K=K(k)$ and
$\beta=\max\(\frac{\log D}{\log n},0\)$.
 
There exists some real number $A_0=A_0(k)>1$ such that for all
classes of functions $\Cal F$ which satisfy the above conditions
of Proposition~4 the set of decoupled $U$-statistics determined by the
functions~$f\in\Cal F$ have a good tail behaviour at level~$T$
for some $T\ge A_0$, provided that for all classes of functions $\Cal F$
with such properties the set of decoupled $U$-statistics with kernel
functions $f\in \Cal F$ have a good tail behaviour at level~$T^{4/3}$.}
\medskip
It is not difficult to deduce Proposition~$2'$ from Proposition~4.
Indeed, let us observe that the set of (decoupled) $U$-statistics
determined by a class of functions $\Cal F$ satisfying the conditions of
Proposition~4 has a good tail-behaviour at level
$T_0=\sigma^{-(k+1)}$, since the probability at the left-hand side
of (3.11) equals zero for $x>\sigma^{-(k+1)}$. Then we get from
Proposition~4 by induction with respect to the number $j$ that all
sets of $U$-statistics $\bar I_{n,k}(f)$, $f\in\Cal F$, with a class of
functions $\Cal F$ satisfying the
conditions of Proposition~4 have a good tail-behaviour also for $T\ge
T_0^{(3/4)^j} =\sigma^{-(3/4)^j(k+1)}$ for all $j=1,2,\dots$ such that
$\sigma^{-(3/4)^j(k+1)}\ge A_0$. This implies that if a class of
functions $\Cal F$ satisfies the conditions of Proposition~4, then
the set of $U$-statistics determined by this class of functions has
a good tail-behaviour at level $T=A_0^{4/3}$, i.e. at a level which
depends only on the order $k$ of the (decoupled) $U$-statistics.
This result implies Proposition~$2'$, only we have to apply it not
directly for the class of functions~$\Cal F$ appearing in
Proposition~$2'$, but these functions have to be multiplied by a
sufficiently small positive number depending only on~$k$.
 
Thus to complete the proof of Theorem~2 it is enough to prove
Proposition~4. I describe its proof in the special case $k=1$ in
the next section. This case is considered separately, because it
may help to understand the ideas of the proof in the general case.
 
The main difficulty in the proof of Proposition 4 is related to a
symmetrization procedure which is an essential part of the proof.
I want to apply some randomization with the help of a
symmetrization argument, and this requires a special justification.
It is not difficult to justify the right for this randomization in
the case $k=1$, when it simply follows from Lemma~3 and a (simple)
estimation of the variance of an appropriate $U$-statistic, but it
becomes hard for~$k\ge2$. In this case we have to give a good estimate
on certain conditional variances of some (decoupled) $U$-statistics
with respect to some appropriate conditions. To overcome this
difficulty I formulate a result in Proposition~5 and prove
Propositions~4 and~5 simultaneously. Their proof follows the
following line. First Proposition~4 and Proposition~5 will be proved
for $k=1$. Then, if Propositions~4 and~5 are already proven for
all $k'<k$, then first I prove Proposition~4 for $k$, and
then Proposition~5 for the same~$k$. Proposition~5 has a structure
similar to that of Proposition~4. Before its formulation I
introduce the following definition.
 
\medskip\noindent
{\bf Definition of good tail behaviour for a class of integrals of
$U$-statistics.} {\it Let us have a product space $(X^k\times
Y,\Cal X^k\times\Cal Y)$ with some product measure $\mu^k\times\rho$,
where $(X^k,\Cal X^k,\mu^k)$ is the $k$-fold product of some
probability space $(X,\Cal X,\mu)$, and $(Y,\Cal Y,\rho)$ is some
other probability space. Fix some positive integer~$n$ and positive
number $\sigma>0$, and consider some class $\Cal F$ of functions
$f(x_1,\dots,x_k,y)$ on the product space $(X^k\times Y,\Cal
X^k\times\Cal Y,\mu^k\times\rho)$. Take $k$ independent copies
$\xi_{1,s},\dots,\xi_{n,s}$, $1\le s\le k$, of a sequence of
independent, $\mu$-distributed random variables $\xi_1,\dots,\xi_n$.
For all $f\in\Cal F$ and $y\in Y$ let us define the decoupled
$U$-statistics $\bar I_{n,k}(f,y)$ by means of these random variables
$\xi_{1,s},\dots,\xi_{n,s}$, $1\le s\le k$, and the kernel function
$f_y(x_1,\dots,x_k)=f(x_1,\dots,x_k,y)$ in formula~(3.1).
Define with the help of these $U$-statistics $\bar I_{n,k}(f,y)$ the
random integrals
$$
H_{n,k}(f)=\int \bar I_{n,k}(f,y)^2\rho(\,dy), \quad f\in\Cal F.
\tag3.12
$$
Choose some real number $T>0$. We say that the set of random
integrals $H_{n,k}(f)$, $f\in\Cal F$, have a good tail behaviour at
level $T$ if
$$
P\biggl(\sup_{f\in\Cal F} n^{-k}H_{n,k}(f)\ge A^2
n^k\sigma^{2k+2}\biggr) \le \exp\left\{-A^{1/(2k+1)}n\sigma^2 \right\}
\quad \text{for } A\ge T. \tag3.13
$$
}
\medskip\noindent
{\bf Proposition 5.} {\it Fix some positive integer $n$ and real
number $0<\sigma\le2^{-(k+1)}$, and let us have a product space
$(X^k\times Y,\Cal X^k\times\Cal Y)$ with some product measure
$\mu^k\times\rho$, where $(X^k,\Cal X^k,\mu^k)$ is the $k$-fold
product of some probability space $(X,\Cal X,\mu)$, and $(Y,\Cal
Y,\rho)$ is some other probability space. Let us have a
countable $L_2$-dense class $\Cal F$ of canonical functions
$f(x_1,\dots,x_k,y)$ on the product space $(X^k\times Y,\Cal
X^k\times\Cal Y,\mu^k\times\rho)$ with some exponent $L\ge1$ and
parameter $D>0$. Let us also assume that the functions
$f\in \Cal F$ satisfy the conditions
$$
\supp_{x_j\in X, 1\le j\le k, y\in Y}|f(x_1,\dots,x_k,y)|\le
2^{-(k+1)}
$$
and
$$
\int f^2(x_1,\dots,x_k,y)\mu(\,dx_1)\dots\mu(\,dx_k)\rho(\,dy)\le
\sigma^2 \quad  \text{for all } f\in \Cal F.
$$
Let the inequality $n\sigma^2>K(L+\beta)\log n$ hold with a
sufficiently large fixed number $K=K(k)$.
 
There exists some number $A_0=A_0(k)>1$ such that for all
classes of functions $\Cal F$ which satisfy the conditions of
Proposition~5 the random integrals $H_{n,k}(f)$, $f\in\Cal F$,
defined in (3.9) have a good tail behaviour at level~$T$, provided
that the random integrals $H_{n,k}(f)$, $f\in\Cal F$, of all
classes $\Cal F$ with such properties have a good tail
behaviour at level~ $T^{(2k+1)/2k}$.} \medskip
 
Similarly to the argument formulated after Proposition~4 an
inductive procedure yields the following corollary of Proposition~5.
\medskip\noindent
{\bf Corollary of Proposition 5.} {\it If the class of functions
$\Cal F$ satisfies the conditions of Proposition~5, then there
exists a constant $\bar A_0=\bar A_0(k)>0$ depending only on $k$
such that the integrals $H_{n,k}(f)$ determined by the class of
functions $\Cal F$ have a good tail behaviour at level $\bar A_0$.}
 
\beginsection 4. The proof of Proposition 4 in the case $k=1$
 
In this section Proposition~4 is proved in the special case $k=1$.
In this case we have to show that
$$
P\(\frac1{\sqrt n}\supp_{f\in\Cal F}\left|\summ_{j=1}^n
f(\xi_j)\right| \ge A n^{1/2}\sigma^{2}\) \le e^{-A^{1/2}
n\sigma^2} \quad \text{if } A\ge T
\tag4.1
$$
if we know the same estimate for $A>T^{4/3}$ and all classes of
functions satisfying the conditions of Proposition~4. This statement
will be proved by means of the following symmetrization argument.
\medskip\noindent
{\bf Lemma 4.} {\it Let the class of functions $\Cal F$ satisfy the
conditions of Proposition~4 for $k=1$. Let $\e_1,\dots,\e_n$ be a
sequence of independent random variables,
$P(\e_j=1)=P(\e_j=-1)=\frac12$, independent also of the
$\mu$ distributed random variables $\xi_1,\dots,\xi_n$. Then
$$
\aligned
&P\(\frac1{\sqrt n}\supp_{f\in\Cal F}\left|\summ_{j=1}^n
f(\xi_j)\right| \ge A
n^{1/2}\sigma^{2}\) \\
&\qquad \le 4P\(\frac1{\sqrt n}\supp_{f\in\Cal F}\left|\summ_{j=1}^n
\e_jf(\xi_j)\right| \ge \frac A3
n^{1/2}\sigma^{2}\) \quad\text{if } A\ge T.
\endaligned \tag4.2
$$
}\medskip
 
There are several similar results in the literature. Lemma~4 follows
simply from Part b) of Lemma~2.7 in~[7] with the choice
$t=An\sigma^2$. (The quantity $\alpha$ in this result agrees with
our $\sigma$.) It is enough to check that $t\ge 2^{1/2}n^{1/2}\sigma$
and $(t-2^{1/2}n^{1/2}\sigma)/2\ge\frac A3n\sigma^2$ if
$A\ge T\ge A_0$  is chosen sufficiently large, since under the
conditions of Proposition~4 (if the parameter~$K$ is
sufficiently large in Proposition~4) $n\sigma^2\ge1$.
 
To prove Proposition~4 for $k=1$  let us investigate the conditional
probability
$$
P(f,A|\xi_1,\dots,\xi_n)=
P\(\left.\frac1{\sqrt n}\left|\summ_{j=1}^n
\e_jf(\xi_j)\right| \ge \frac
A6\sqrt n\sigma^2\right|\xi_1,\dots,\xi_n\)
$$
for all functions $f\in\Cal F$, $A\ge T$ and values
$(\xi_1,\dots,\xi_n)$. By Lemma~2 (with $k=1$) we can write
$$
P(f,A|\xi_1,\dots,\xi_n)\le C\exp\left\{-\frac{\frac B{36}
A^2 n\sigma^4}{S^2(f,\xi_1,\dots,\xi_n)}\right\} \tag4.3
$$
with
$$
S^2(f,x_1,\dots,x_n)=\frac1n\sum_{j=1}^n f^2(x_j), \quad f\in \Cal F.
$$
Let us introduce the set
$$
H=H(A)=\left\{(x_1,\dots,x_n)\colon \sup_{f\in\Cal F}
S^2(f,x_1,\dots,x_n)\ge \(1+A^{4/3}\)\sigma^2\right\}. \tag4.4
$$
I claim that
$$
P((\xi_1,\dots,\xi_n)\in H)\le e^{-A^{2/3} n\sigma^2}\quad\text{ if }
A\ge T. \tag4.5
$$
To prove relation (4.5) let us consider the functions
$\bar f=\bar f(f)$ for all $f\in \Cal F$ defined by the formula
$\bar f(x)=f^2(x)-\int f^2(x)\mu(\,dx)$, and introduce the class
of functions $\Cal F'=\{\bar f(f)\colon f\in\Cal F\}$. Let us show that
the class of functions $\Cal F'$ satisfies the conditions of
Proposition~4. By the assumption of Proposition~4 this implies that
the estimate~(3.11) with $k=1$, i.e. the estimate~(4.1) holds for
the class of functions $\Cal F'$ if the condition $A\ge T$ is
replaced by $A\ge T^{4/3}$ in it.
 
Relation $\int \bar f(x)\mu(\,dx)=0$ clearly holds. (In the case
$k=1$ this means that $\bar f$ is a canonical function.) The condition
$\sup|\bar f(x)|\le\frac 18<\frac14$ also holds if $\sup|f(x)|\le
\frac14$, and $\int\bar f^2(x)\mu(\,dx)\le \int f^4(x)\mu(\,dx)\le
\frac14\int f^2(x)\,\mu(\,dx)\le\frac{\sigma^2}4<\sigma^2$ if $f\in
\Cal F$. It remained to show that $\Cal F'$ is an $L_2$-dense class
with exponent $L$ and parameter $D$.
 
To show this observe that $\int (\bar f(x)-\bar g(x))^2\rho(\,dx)\le
2\int(f^2(x)-g^2(x))^2\rho(\,dx)+
2\int(f^2(x)-g^2(x))^2\mu(\,dx)\le2 (\supp (|f(x)|+|g(x)|)^2
\(\int (f(x)-g(x))^2(\rho(\,dx)+\mu(\,dx)\)\le  \int
(f(x)-g(x))^2\bar\rho(\,dx)$ for all $f, g\in\Cal F$, $\bar f=\bar
f(f)$, $\bar g=\bar g(g)$ and probability measure $\rho$, where
$\bar\rho=\frac{\rho+\mu}2$. This means that if $\{f_1,\dots,f_m\}$
is an $\e$-dense subset of $\Cal F$ in the space $L_2(X,\Cal
X,\bar\rho)$, then $\{\bar f_1,\dots,\bar f_m\}$ is an $\e$-dense
subset of $\Cal F'$ in the space $L_2(X,\Cal X,\rho)$, and
not only $\Cal F$, but also $\Cal F'$ is an $L_2$-dense class with
exponent $L$ and parameter $D$.
 
We get, by applying formula (4.1) for the number $A^{4/3}\ge
T^{4/3}$ and the class of functions $\Cal F'$ that
$$
\align
P((\xi_1,\dots,\xi_n)\in H)&=P\(\sup_{f\in\Cal F} \(\frac1n \sum_{j=1}^n
\bar f(\xi_j) +\frac1n \sum_{j=1}^n E f^2(\xi_j)\)
\ge \(1+A^{4/3}\)\sigma^2\)\\
&\le P\(\sup_{f\in\Cal F} \frac1{\sqrt n} \sum_{j=1}^n \bar f(\xi_j)
\ge A^{4/3}n^{1/2}\sigma^2\) \le e^{-A^{2/3} n\sigma^2},
\endalign
$$
i.e. relation (4.5) holds.
 
Formula (4.3) and the definition (4.4) of the set $H$ yield the estimate
$$
P(f,A|\xi_1,\dots,\xi_n)\le Ce^{- B A^{2/3} n\sigma^2/40} \quad
\text{if }(\xi_1,\dots,\xi_n)\notin H \tag4.6
$$
for all $f\in \Cal F$ and $A\ge T$ for the conditional
probability $P(f,A|\xi_1,\dots,\xi_n)$.
Let us introduce the conditional probability
$$
P(\Cal F,A|\xi_1,\dots,\xi_n)=
P\(\left.\sup_{f\in \Cal F} \frac1{\sqrt n}\left|\summ_{j=1}^n
\e_jf(\xi_j)\right| \ge \frac
A3\sqrt n\sigma^2\right|\xi_1,\dots,\xi_n\)
$$
for all $(\xi_1,\dots,\xi_n)$ and $A\ge T$. We shall
estimate this conditional probability with the help of relation (4.6)
if $(\xi_1,\dots,\xi_n) \notin H$. Given some set of $n$~points
$(x_1,\dots,x_n)$ in the space $(X,\Cal X)$ let us introduce the
measure $\nu=\nu(x_1,\dots,x_n)$ on $(X,\Cal X)$ in such a way that
$\nu$ is concentrated in the points $x_1,\dots,x_n$, and
$\nu(\{x_j\})=\frac1n$. If $\int f^2(x)\nu(\,dx)\le\delta^2$ for a
function $f$, then $\left|\frac1{\sqrt n}\summ_{j=1}^n
\e_jf(x_j)\right|\le n^{1/2}\int|f(x)|\nu(\,dx)\le n^{1/2}\delta$.
Since we have assumed that $n\sigma^2\ge1$, this estimate implies
that if $f$ and $g$ are two functions such that $\int
(f-g)^2\nu(\,dx)\le \delta^2$ with $\delta=\frac A{6n}$,
then $\left|\frac1{\sqrt n}\summ_{j=1}^n \e_jf(x_j)-
\frac1{\sqrt n}\summ_{j=1}^n \e_jg(x_j)\right|\le\frac A{6\sqrt n}
\le\frac A6 \sqrt n\sigma^2$.
 
Given some (random) point $(\xi_1,\dots,\xi_n)\in H$ let us consider
the measure $\nu=\nu(\xi_1,\dots,\xi_n)$ corresponding to it, and
choose a $\bar\delta$-dense subset $\{f_1,\dots,f_m\}$ of $\Cal F$ in
the space $L_2(X,\Cal X,\nu)$ with $\bar\delta=\frac1{6n}\le\delta=
\frac A{6n}$, whose cardinality $m$ satisfies the inequality $m\le
D\bar\delta^{-L}$. This is possible because of the $L_2$-dense
property of the class~$\Cal F$. (This is the point where the
$L_2$-dense property of the class of functions $\Cal F$ is exploited
in its full strength.) The above facts imply that $P(\Cal
F,A|\xi_1,\dots,\xi_n)\le\summ_{l=1}^m P(f_l,A|\xi_1,\dots,\xi_n)$ with
these functions $f_1,\dots,f_m$. Hence relation (4.6) yields that
$$
P(\Cal F,A|\xi_1,\dots,\xi_n)\le CD(6n)^Le^{- B
A^{2/3} n\sigma^2/40} \quad \text{if }(\xi_1,\dots,\xi_n)\notin H
\text{ and } A\ge T.
$$
This inequality together with Lemma~4 and estimate~(4.5) imply that
$$
\aligned
&P\(\frac1{\sqrt n}\supp_{f\in\Cal F}\left|\summ_{j=1}^n
f(\xi_j)\right| \ge A n^{1/2}\sigma^{2}\)
\le 4P\(\frac1{\sqrt n}\supp_{f\in\Cal F}\left|\summ_{j=1}^n
\e_jf(\xi_j)\right| \ge \frac A3 n^{1/2}\sigma^{2}\)\\
&\qquad \le 4CD(6n)^Le^{- B A^{2/3}n\sigma^2/40}
+4e^{-A^{2/3}n\sigma^2} \quad \text{if } A\ge T.
\endaligned \tag4.7
$$
Since we have a better power of $A$ in the exponent at the
right-hand side of formula (4.7) than we need, the relation
$n\sigma^2\ge K(L+\beta)\log n$ holds, and we have the right to
choose the constants $K$ and $A_0$, $A\ge A_0$, sufficiently large,
it is not difficult to deduce relation (4.1) from relation (4.7).
Indeed, the expression in the exponent at the right-hand side of
(4.7) satisfies the inequality $\frac B{40} A^{2/3} n\sigma^2\ge
A^{1/2} n\sigma^2+K(L+\beta)\log n$ if $A_0$ is sufficiently
large, and
$$
\align
P&\(\frac1{\sqrt n}\supp_{f\in\Cal F}\left|\summ_{j=1}^n
f(\xi_j)\right| \ge An^{1/2}\sigma^{2}\)\\
&\qquad \le 4C(6n)^{\beta+L}e^{-K} n^{-K(L+\beta)}
e^{-A^{1/2} n\sigma^2}+4e^{-A^{2/3}n\sigma^2}\le e^{-A^{1/2}n\sigma^2}
\endalign
$$
if $A\ge T$, and the constants $A_0$ and $K$ are chosen sufficiently
large.
 
\beginsection 5. The symmetrization argument
 
\medskip\noindent
In the proof of Propositions~4 and~5 we need two symmetrization
results for all $k\ge1$ which play the same role as Lemma~4 in the
case $k=1$. These results are described in Lemmas~5A and~5B. In
this section these results are formulated and proved. The proofs
go by induction with respect to $k$. During the proof of Propositions~4
and~5 for~$k$ we may assume that they hold for $k'<k$.
\medskip\noindent
{\bf Lemma 5A.} {\it Let $\Cal F$ be a class of functions on the
space $(X^k,\Cal X^k)$ which satisfies the conditions of Proposition~4
with some probability measure $\mu$. Let us have $k$ independent
copies $\xi_{1,s},\dots,\xi_{n,s}$, $1\le s\le k$, of a sequence of
independent $\mu$ distributed random variables $\xi_1,\dots,\xi_n$,
and a sequence of independent random variables $\e=(\e_1,\dots,\e_n)$,
$P(\e_s=1)=P(\e_s=-1)=\frac12$, which is independent also of the
random variables $\xi_{j,s}$, $1\le j\le n$, $1\le s\le k$. Consider
the decoupled $U$-statistics $\bar I_{n,k}(f)$, $f\in\Cal F$,
defined from these random variables by formula~(3.1) and their
randomized version
$$
\bar I_{n,k}^{\e}(f)=\frac1{k!}\summ\Sb 1\le j_s\le n,\; s=1,\dots,
k\\ j_s\neq j_{s'} \text{ if } s\neq s'\endSb
\e_{j_1}\cdots\e_{j_k}f\(\xi_{j_1,1},\dots,
\xi_{j_k,k}\),  \quad f\in\Cal F. \tag5.1
$$
There exists some constant $A_0=A_0(k)$ such that the inequality
$$
\aligned
P\(\sup_{f\in\Cal F} n^{-k/2}\left|\bar
I_{n,k}(f)\right|>An^{k/2}\sigma^{k+1}\)&<
2^{k+1}P\(\sup_{f\in\Cal F} \left|\bar I_{n,k}^{\e}(f)\right|
>2^{-(k+1)}A n^k\sigma^{k+1}\)\\
&\qquad+2^kn^{k-1}e^{-A^{1/(2k-1)} n\sigma^2/k}
\endaligned \tag 5.2
$$
holds for all $A\ge A_0$.}
\medskip
Before formulating Lemma 5B needed in the proof of Proposition~5
I introduce some notations. Some of them will be needed later.
 
Let us consider a set $\Cal F$ of functions
$f(x_1,\dots,x_k,y)\in \Cal F$ on a space $(X^k\times Y, \Cal X^k
\times \Cal Y,\mu^k\times\rho)$ which satisfies the conditions of
Proposition~5. Let us choose $2k$ independent copies
$\xi_{1,s}^{(1)},\dots,\xi_{n,s}^{(1)}$,
$\xi_{1,s}^{(-1)},\dots,\xi_{n,s}^{(-1)}$, $1\le s\le k$, of a
sequence of independent $\mu$ distributed random variables
$\xi_1,\dots,\xi_n$ together with a sequence of independent random
variables $(\e_1,\dots,\e_n)$, $P(\e_s=1)=P(\e_s=-1)=\frac12$, $1\le
s\le n$, which are independent of them. For all subsets
$V\subset\{1,\dots,k\}$
of the set $\{1,\dots,k\}$ let $|V|$ denote the cardinality of this set,
and define for all functions $f(x_1,\dots,x_k,y)\in \Cal F$ and
$V\subset\{1,\dots,k\}$ the decoupled $U$-statistics
$$
\bar I_{n,k}^V(f,y)=\frac1{k!}\summ\Sb 1\le j_s\le n,\; s=1,\dots, k\\
j_s\neq j_{s'} \text{ if } s\neq s'\endSb
f\(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y\),\quad
f\in\Cal F, \tag5.3
$$
where $\delta_s=\pm1$, $1\le s\le k$, $\delta_s=1$ if $s\in V$,
and $\delta_s=-1$ if $s\notin V$, together with the random variables
$$
H_{n,k}^V(f)=\int \bar I_{n,k}^V(f,y)^2\rho(\,dy), \quad f\in\Cal
F. \tag$5.3'$
$$
Put
$$
\bar I_{n,k}(f,y)=\bar I_{n,k}^{\{1,\dots,k\}}(f,y),\quad
H_{n,k}(f)=H_{n,k}^{\{1,\dots,k\}}(f), \tag$5.3''$
$$
i.e. these random variables agree with those defined in (5.3) and
$(5.3')$ with the choice $V=\{1,\dots,k\}$.
 
Let us also define the `randomized version' of the random variables
$\bar I_{n,k}^V(f,y)$ and $H_{n,k}^V(f)$ as
$$
\bar I_{n,k}^{(V,\e)}(f,y)=\frac1{k!}\summ\Sb 1\le j_s\le n,\;
s=1,\dots, k\\ j_s\neq j_{s'} \text{ if } s\neq s'\endSb
\e_{j_1}\cdots\e_{j_k}f\(\xi_{j_1,1}^{(\delta_1)},\dots,
\xi_{j_k,k}^{(\delta_k)},y\),\quad f\in\Cal F, \tag5.4
$$
where $\delta_s=1$ if $s\in V$, and $\delta_s=-1$ if $s\notin V$, and
$$
H_{n,k}^{(V,\e)}(f)=\int \bar I_{n,k}^{(V,\e)}(f,y)^2\rho(\,dy)
,\quad f\in\Cal F. \tag$5.4'$
$$
 
Let us also introduce the random variables
$$
\bar W(f)=\int\[\sum_{V\subset \{1,\dots,k\}} (-1)^{|V|}\bar
I_{n,k}^{(V,\e)}(f,y)\]^2\rho(\,dy), \quad f\in\Cal F. \tag5.5
$$
Now I formulate the symmetrization result applied
in the proof of Proposition~5.
\medskip\noindent
{\bf Lemma 5B.} {\it Let $\Cal F$ be a set of functions on
$(X^k\times Y,\Cal X^k\times\Cal Y)$ which satisfies the conditions
of Proposition~5 with some probability measure $\mu^k\times\rho$.
Let us have $2k$ independent copies
$\xi_{1,s}^{\pm1},\dots,\xi_{n,s}^{\pm1}$, $1\le s\le k$, of a
sequence of independent $\mu$ distributed random variables
together with a sequence of independent random
variables $\e_1,\dots,\e_n$, $P(\e_s=1)=P(\e_s=-1)=\frac12$, $1\le s\le
n$, which is independent of them.
 
There exists some $A_0=A_0(k)$ such that if the integrals
$H_{n,k}(f)$, $f\in\Cal F$, determined by this class of functions
$\Cal F$ have a good tail behaviour at level $T^{(2k+1)/2k}$ for
some $T\ge A_0$, (this property was defined at the end of
Section~3), then the inequality
$$
\aligned
P\(\sup_{f\in\Cal F} H_{n,k}(f)>A^2n^{2k}\sigma^{2(k+1)}\)
&<2P\(\sup_{f\in\Cal F} \left|\bar W(f)\right|
>\frac{A^2}2 n^{2k}\sigma^{2(k+1)}\)\\
&\qquad+2^{2k+1}n^{k-1}e^{-A^{1/2k} n\sigma^2/k}
\endaligned \tag 5.6
$$
holds with the random variables $H_{n,k}(f)$ and $\bar W(f)$ defined
in formulas $(5.3'')$ and (5.5) for all $A\ge T$.}
\medskip
 
Let us observe that in the symmetrization argument of Lemma~5B we have
applied the randomization $\bar I_{n,k}^{(V,\e)}(f,y)$ of  $\bar
I_{n,k}^{(V)}(f,y)$, (compare formulas (5.3) and (5.4)), and compared
the integral of the square of the random function $\bar I_{n,k}(f,y)$
with the integral of the square of a linear combination of the random
functions $\bar I_{n,k}^{(V,\e)}(f,y)$. After this integration the
effect of the `randomizing factors' $\e_j$ will be weaker.
Nevertheless, also such an estimate will be sufficient for us. But
the effect of this symmetrization procedure has to be followed more
carefully. Hence a corollary of Lemma~5B will be presented which can
be better applied than the original lemma. We get it by rewriting the
random variable $\bar W(f)$ defined in (5.5) in another form with the
help of some diagrams introduced below.
 
Let $\Cal G=\Cal G(k)$ denote the set of all diagrams consisting of
two rows such that both rows are the set $\{1,\dots,k\}$ and the
diagrams of $\Cal G$ contain some edges $(l_1,l_1')$,\dots,
$(l_s,l_s')$, $0\le s\le k$ connecting some points (vertices) of
the first row with some points of the second row. The
vertices $l_1,\dots,l_s$ in the first row are all different, and
the same relation holds also for the vertices $l_1',\dots,l_s'$ in
the second row. For each diagram $G\in\Cal G$ let us define
$e(G)=\{(l_1,l_1')\dots,(l_s,l_s')\}$, the set of its edges,
$v_1(G)=\{l_1,\dots,l_s\}$, the set of vertices in the first
row and $v_2(G)=\{l_1',\dots,l_s'\}$, the set of vertices in the
second row of $\Cal G$ from which an edge starts.
 
Given some diagram $G\in \Cal G$ and two sets
$V_1,V_2\subset\{1,\dots,k\}$, we define with the help of the
random variables $\xi_{s,1}^{(1)},\dots,\xi_{s,n}^{(1)}$,
$\xi_{s,1}^{(-1)},\dots,\xi_{s,n}^{(-1)}$, $1\le s\le k$, and
$\e=(\e_1,\dots,\e_n)$ taking part in the definition of the
expressions $\bar W(f)$, $f\in\Cal F$, the random variables
$H_{n,k}(f|G,V_1,V_2)$:
$$ \allowdisplaybreaks
\align
H_{n,k}(f|G,V_1,V_2)&=\sum\Sb(j_1,\dots,j_k,\;j'_1,\dots,j'_k) \\
1\le j_s\le n,\, j_s\neq j_{s'} \text{ if }s\neq s',\,1\le s,s'\le k,\\
1\le j'_s\le n,\, j'_s\neq j'_{s'}\text { if } s\neq s',\,1\le s,s'\le
k,\\ j_s=j'_{s'} \text { if } (s,s')\in e(G),\; j_s\neq j'_{s'} \text
{ if } (s,s')\notin e(G)\endSb  \!\!\!\!\!\!\!\!\!\!\!\!
\prod_{s\in \{1,\dots,k\}\setminus v_1(G)} \!\!\!\! \e_{j_s}
\prod_{s\in \{1,\dots,k\}\setminus v_2(G)} \!\!\!\! \e_{j'_s} \\
&\qquad \frac1{k!^2} \int
f(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y)
f(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}^{(\bar\delta_k)},y)
\rho(\,dy), \quad f\in\Cal F,  \tag5.7
\endalign
$$
where $\delta_s=1$ if $s\in V_1$, $\delta_s=-1$ if $s\notin V_1$,
and $\bar\delta_s=1$ if $s\in V_2$, $\bar\delta_s=-1$ if $s\notin V_2$.
 
With the help of these random variables we can write that
$$
\bar W(f)=\sum_{G\in \Cal G,\, V_1,V_2\subset \{1,\dots,k\}}
(-1)^{|V_1|+|V_2|} H_{n,k}(f|G,V_1,V_2) \quad \text{for all }
f\in\Cal F,
$$
because
$$
\int\bar I_{n,k}^{(V_1,\e)}(f,y)\bar I_{n,k}^{(V_2,\e)}(f,y)\rho(\,dy)
=\sum_{G\in\Cal G} H_{n,k}(f|G,V_1,V_2), \quad \text{for all }
V_1,V_2\subset\{1,\dots,k\}.
$$
 
Since the number of terms in this sum is less than $2^{4k}k!$, it
implies that Lemma~5B has the following corollary:
\medskip\noindent
{\bf Corollary of Lemma 5B.} {\it Let a set of functions $\Cal F$
satisfy the conditions of Proposition~5. Then there exists some
$A_0=A_0(k)$ such that if the integrals
$H_{n,k}(f)$, $f\in\Cal F$, determined by this class of functions
$\Cal F$ have a good tail behaviour at level $T^{(2k+1)/2k}$ for
some $T\ge A_0$, then the inequality
$$
\align
&P\(\sup_{f\in\Cal F} H_{n,k}(f)>A^2n^{2k}\sigma^{2(k+1)}\)\\
&\qquad\qquad\le 2\sum_{G\in \Cal G,\, V_1,V_2\in\{1,\dots,k\}}
P\(\sup_{f\in\Cal F} \left |H_{n,k}(f|G,V_1,V_2)\right|
>\frac{A^2}{2^{4k+1}k!} n^{2k}\sigma^{2(k+1)}\) \\
&\qquad\qquad\qquad +2^{2k+1}n^{k-1}e^{-A^{1/2k} n\sigma^2/k} \tag 5.8
\endalign
$$
holds with the random variables $H_{n,k}(f)$ and $H_{n,k}(f|G,V_1,V_2)$
defined in formulas $(5.3'')$ and (5.7) for all $A\ge T$.}
\medskip
 
The proof of Lemmas 5A and 5B uses the result of the following
Lemma~6 which states that certain random vectors have the same
distribution.
\medskip\noindent
{\bf Lemma 6.} {\it Let $\e=(\e_1,\dots,\e_n)$ be a sequence of
independent random variables, $P(\e_s=1)=P(\e_s=-1)=\frac12$,
$1\le s\le n$, which is independent also of $2k$ fixed independent
copies $\xi_{1,s}^{(1)},\dots,\xi_{n,s}^{(1)}$ and
$\xi_{1,s}^{(-1)},\dots,\xi_{n,s}^{(-1)}$, $1\le s\le k$, of a
sequence $\xi_1,\dots,\xi_n$ of independent $\mu$ distributed random
variables. \medskip
\item{a)} Let $\Cal F$ be a class of functions which satisfies the
conditions of Proposition 4. With the help of the above
random variables introduce the decoupled $U$-statistic
$$
\bar I_{n,k}^V(f)=\frac1{k!}\summ\Sb 1\le j_s\le n,\; s=1,\dots, k\\
j_s\neq j_{s'} \text{ if } s\neq s'\endSb
f\(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)}\),\quad
f\in\Cal F, \tag5.9
$$
for all sets $V\subset\{1,\dots,k\}$ and functions $f\in \Cal F$
together with its `randomized version'
$$
\bar I_{n,k}^{(V,\e)}(f)=\frac1{k!}\summ\Sb 1\le j_s\le n,\;
s=1,\dots, k\\ j_s\neq j_{s'} \text{ if } s\neq s'\endSb
\e_{j_1}\cdots\e_{j_k}f\(\xi_{j_1,1}^{(\delta_1)},\dots,
\xi_{j_k,k}^{(\delta_k)}\),  \quad f\in\Cal F, \tag$5.9'$
$$
where $\delta_s=\pm1$, $1\le s\le k$, $\delta_s=1$ if $s\in V$, and
$\delta_s=-1$ if $s\notin V$.
 
Then the sets of random variables
$$
S(f)=\sum_{V\subset \{1,\dots,k\}} (-1)^{|V|}\bar I_{n,k}^V(f),
\quad f\in\Cal F, \tag5.10
$$
and sets of random variables
$$
\bar S(f)=\sum_{V\subset \{1,\dots,k\}} (-1)^{|V|}\bar
I_{n,k}^{(V,\e)}(f), \quad f\in\Cal F, \tag$5.10'$
$$
have the same joint distribution.
\medskip
\item{b)} Let $\Cal F$ be a class of functions satisfying
Proposition 5. For all functions $f\in \Cal F$ and
$V\subset\{1,\dots,k\}$ consider the decoupled $U$-statistics
$\bar I_{n,k}^V(f,y)$ determined by the random variables
$\xi_{1,s}^{(1)},\dots,\xi_{n,s}^{(1)}$ and
$\xi_{1,s}^{(-1)},\dots,\xi_{n,s}^{(-1)}$, $1\le s\le k$, by
formula (5.3), and define with their help the random variables
$$
W(f)=\int\[\sum_{V\subset \{1,\dots,k\}} (-1)^{|V|}\bar
I_{n,k}^V(f,y)\]^2\rho(\,dy), \quad f\in\Cal F. \tag5.11
$$
Then the random vectors $\{W(f)\colon f\in \Cal F\}$ defined in (5.11)
and $\{\bar W(f)\colon f\in \Cal F\}$ defined in (5.5)
have the same distribution.}
\medskip\noindent
{\it Proof of Lemma 6.} Let us consider Part a) of Lemma~6. I claim
that for all $M\in\{1,\dots,n\}$  the conditional distribution of
the random vector in $(5.10')$ under the condition that $\e_j=1$ if
$j\in M$ and $\e_j=-1$ if $\e_j\in\{1,\dots,n\}\setminus M$ agrees
with the distribution of the vector in (5.10). Since the
distribution of the vector in (5.10) does not change if we exchange
the random variables $\xi_{j,s}^{(1)}$ and $\xi_{j,s}^{(-1)}$ in it for
$j\notin M$, $1\le s\le k$, and do not exchange them otherwise, it is
enough to understand that the random vector we get from the vector
in (5.10) after this transformation agrees with the random vector in
$(5.10')$ if we write $\e_j=1$ for $j\in M$ and $\e_j=-1$ for
$j\notin M$ in it. These random vectors really agree (not only in
distribution) since for all functions $f\in \Cal F$ both vectors
have a component which is the sum of terms of the form
$f(\xi_{j_1,1}^{(\delta_{j_1})},\dots,\xi_{j_k,k}^{(\delta_{j_k})})$,
$\delta_{j_s}=\pm1$, $1\le s\le k$, multiplied with an appropriate
power of $-1$, and this power equals the number of $-1$ components
in the sequence $\delta_{j_1},\dots,\delta_{j_k}$ plus the cardinality
of the set $\{j_1,\dots,j_k\}\cap M$.
 
Part b) of Lemma~6 can be proved in the same way, hence it is omitted.
 
\medskip
Lemma 5A will be proved with the help of part a) of Lemma~6 and the
following Lemma~7A.
\medskip\noindent
{\bf Lemma 7A.} {\it Let us consider a class of functions $\Cal F$
satisfying the conditions of Proposition~4 and the random variables
$\bar I_{n,k}^V(f)$, $f\in\Cal F$, $V\subset\{1,\dots,k\}$, defined
in formula~(5.1). Let $\Cal B=\Cal
B(\xi_{1,s}^{(1)},\dots,\xi_{n,s}^{(1)};\,1\le s\le k)$ denote the
$\sigma$-algebra generated by the random variables
$\xi_{1,s}^{(1)},\dots,\xi_{n,s}^{(1)}$ , $1\le s\le k$, taking part in
the definition of the random variables $\bar I_{n,k}^V(f)$. For all
$V\subset\{1,\dots,k\}$, $V\neq\{1,\dots,k\}$, there exists a number
$A_0=A_0(k)>0$ such that the inequality
$$
P\(\sup_{f\in\Cal F}\left. E\(\bar I_{n,k}^V(f)^2\right|\Cal B\)
> 2^{-(3k+3)}A^2n^{2k}\sigma^{2k+2}\)<
n^{k-1}e^{-A^{1/(2k-1)} n\sigma^2/k}
\tag5.12
$$
holds for all $A\ge A_0$.}
\medskip\noindent
{\it Proof of Lemma 7A.}\/ Let us first consider the case
 $V=\emptyset$. We have $\left.E\(\bar I_{n,k}^\emptyset(f)^2\right|
\Cal B\) =E\(\bar I_{n,k}^\emptyset(f)^2\)\le\frac{n!}{k!}\sigma^2
\le n^{2k}\sigma^{2k+2}$ for all $f\in\Cal F$. In the above
calculation we exploited that the functions $f\in\Cal F$ are
canonical, and this implies certain orthogonalities, and beside
this the inequality $n\sigma^2\ge1$ holds. The above relation
implies inequality (5.12) for $V=\emptyset$ for all $\oo\in\Omega$
if the number $A_0$ is chosen sufficiently large.
 
To avoid some complications in the notation let us restrict our
attention to the sets $V=\{1,\dots,u\}$, $1\le u<k$, and prove
relation (5.12) for such sets. For this goal let us introduce the
random variables
$$
\bar I_{n,k}^V(f,j_{u+1},\dots,j_k)=\frac1{k!}\summ\Sb 1\le j_s\le
n,\; s=1,\dots, u\\ j_s\neq j_{s'} \text{ if } s\neq s',\;1\le
s,s'\le k\endSb
f\(\xi_{j_1,1}^{(1)},\dots,\xi_{j_u,u}^{(1)},\xi_{j_{u+1},u+1}^{(-1)},
\dots, \xi_{j_k,k}^{(-1)}\),
$$
for all $f\in\Cal F$, i.e. we fix some indices $j_{u+1},\dots,j_k$,
$1\le j_s\le n$, $u+1\le s\le k$, $j_s\neq j_{s'}$ if $s\neq s'$, and
sum up only those terms in the sum defining $\bar I_{n,k}^V(f)$
which contain $\xi_{j_{u+1},u+1}^{(-1)},\dots, \xi_{j_k,k}^{(-1)}$ in
their last $k-u$ coordinates. Then we can write
$$
\aligned
\left.E\(\bar I_{n,k}^V(f)^2\right|\Cal B\)&=
\left.E\(\(\summ\Sb 1\le j_s\le n\; s=u+1,\dots,k\\ j_s\neq j_{s'}\text
{if } s\neq s'\endSb
\bar I_{n,k}^V(f,j_{u+1},\dots,j_{u_k})\)^2\right|\Cal B\) \\
&=\summ\Sb 1\le j_s\le n\; s=u+1,\dots,k\\ j_s\neq j_{s'}\text
{if } s\neq s'\endSb
\left.E\(\bar I_{n,k}^V(f,j_{u+1},\dots,j_{u_k})^2\right|\Cal B\).
\endaligned \tag5.13
$$
The last relation follows from the identity
$$
\left.E\(\bar I_{n,k}^V(f,j_{u+1},\dots,j_{u_k})
\bar I_{n,k}^V(f,j'_{u+1},\dots,j'_{u_k})\right|\Cal B\)=0
$$
if $(j_{u+1},\dots,j_k)\neq(j'_{u+1},\dots,j'_k)$, which relation
holds, since $f$ is a canonical function.
 
It follows from relation (5.13) that
$$
\aligned
&\left\{\oo\colon\sup_{f\in\Cal F}\left.E\(\bar I_{n,k}^V(f)^2\right|
\Cal B\)(\oo) > 2^{-(3k+3)}A^2n^{2k}\sigma^{2k+2}  \right\}\\
&\qquad \subset \bigcup \Sb 1\le j_s\le n\; s=u+1,\dots,k\\
j_s\neq j_{s'} \text {if } s\neq s'\endSb
\left\{\oo\colon \sup_{f\in\Cal F}\left. E\(\bar
I_{n,k}^V(f,j_{u+1},\dots,j_{u_k})^2\right|\Cal
B\)(\oo)>\frac{A^2n^{2k}\sigma^{2k+2}}{2^{(3k+3)}n^{k-u}} \right\}.
\endaligned
\tag5.14
$$
The probability of the events in the union at the right-hand side
of (5.14) can be estimated with the help of the corollary of
Proposition~5 with parameter $u<k$ instead of $k$. (We may assume
that Proposition~5 holds for $u<k$.) This corollary yields that
$$
P\(\sup_{f\in\Cal F}\left. E\(\bar
I_{n,k}^V(f,j_{u+1},\dots,j_{u_k} )^2\right|\Cal
B\)>\frac {A^2\sigma^{2k+2}n^{k+u}} {2^{(3k+3)}}\)\le
e^{-A^{-1/(2u+1)}(n-u)\sigma^2}. \tag5.15
$$
Indeed, the expression $\left. E\(\bar
I_{n,k}^V(f,j_{u+1},\dots,j_{u_k} )^2\right|\Cal B\)$
can be calculated in the following way: Take the decoupled
$U$-statistic
$$
\bar I_{n,k}^V(f,x_{u+1},\dots,x_k)=\frac1{k!}\summ\Sb
j_s\in\{1,\dots,n\}\setminus\{j_{u+1},\dots,j_k\},\\
s=1,\dots, u,\; j_s\neq j_{s'} \text{ if } s\neq s'\endSb
f\(\xi_{j_1,1}^{(1)},\dots,\xi_{j_u,u}^{(1)},x_{u+1},\dots,x_k\)
\tag5.16
$$
of order $u$ with sample size $n-k+u$, and integrate the square of
this function with respect to the variables $x_{u+1},\dots,x_k$ by
the measure $\mu^{k-u}$. The expression  at the left-hand side
of (5.15) can be bounded by means of Proposition 5 if we apply it
for our class of functions $\Cal F$  considering them as functions
on $(X^u\times Y, \Cal X^u\times \Cal Y, \mu^u\times\rho)$ with
$(Y,\Cal Y,\rho)=(X^{k-u},\Cal X^{k-u},\mu^{k-u})$. (A small
inaccuracy was committed in the above statement because to
define the expression in (5.16) as a $U$-statistic we should have
divided by $u!$ instead of $k!$. But this causes no real problem.)
 
We get inequality (5.15) from Proposition~5 by replacing the level
$\frac {A^2\sigma^{2k+2}n^{k+u}} {2^{(3k+3)}}$ in the
probability at the left-hand side by $A^2(n-u)^{2u} \sigma^{2u+2}
<\frac {A^2\sigma^{2k+2}n^{k+u}}{2^{(2k+2)}}$. The last
inequality really holds if the constant $K$ is chosen sufficiently
large in the condition $n\sigma^2>K\log n$ of Proposition~4.
 
Relations (5.14) and (5.15) imply that
$$
P\(\sup_{f\in\Cal F}\left. E\(\bar I_{n,k}^V(f)^2\right|
\Cal B\)(\oo) > 2^{-(3k+3)}A^2n^{2k}\sigma^{2k+2} \)\le
n^{k-u}e^{-A^{-1/(2u+1)}(n-u)\sigma^2},
$$
and $u\le k-1$. Hence also inequality (5.12) holds.
\medskip
Now I prove Lemma~5A.
\medskip\noindent
{\it Proof of Lemma 5A.} I show with the help of Lemma 3 and
Lemma~7A that
$$
\aligned
P\(\sup_{f\in\Cal F} n^{-k/2}\left|\bar
I_{n,k}(f)\right|>An^{k/2}\sigma^{k+1}\)&<
2P\(\sup_{f\in\Cal F} |S(f)|>\frac A2n^k\sigma^{k+1}\)\\
&\qquad +2^kn^{k-1}e^{-A^{1/(2k-1)} n\sigma^2/k}
\endaligned \tag5.17
$$
with the function $S(f)$ defined in (5.10). To prove relation
(5.17) introduce the random variables
$Z(f)=(-1)^{k}\bar I_{n,k}^{\{1,\dots,k\}}(f)$
and $\bar Z(f)=\summ_{V\subset \{1,\dots,k\},\,
V\neq\{1,\dots,k\}}(-1)^{|V|+1}\bar I_{n,k}^V(f)$ for all
$f\in\Cal F$, the $\sigma$-algebra $\Cal B$ considered in Lemma~7A
and the set
$$
B=\bigcap\Sb V\subset\{1,\dots,k\}\\V\neq\{1,\dots,k\}\endSb
\left\{\oo\colon \sup_{f\in\Cal F}\left.E\(\bar I_{n,k}^V(f)^2\right|
\Cal B\)(\oo) \le 2^{-(3k+3)}A^2n^{2k}\sigma^{2k+2}\right\}.
$$
 
Observe that $S(f)=Z(f)-\bar Z(f)$, $f\in\Cal F$, $B\in\Cal B$,
and by Lemma~7A  the inequality $1-P(B)\le2^kn^{k-1}
e^{-A^{1/(2k-1)} n\sigma^2/k}$ holds. Hence to prove relation
(5.17) as a consequence of Lemma~3 it is enough to show that
$$
\left.P\(|\bar Z(f)|>\frac A2n^k\sigma^{k+1}\right|\Cal
B\)(\oo)\le\frac12 \quad \text{ for all }f\in\Cal F \quad \text {if }
\oo\in\Cal B. \tag5.18
$$
But $P\(|\bar I_{n,k}^V(f)|>2^{-(k+1)} An^k\sigma^{k+1}|\Cal
F\)(\oo)\le 2^{-(k+1)}$ for all $f\in \Cal F$, $V\subset\{1,\dots,k\}$,
$V\neq\{1,\dots,k\}$ if $\oo\in B$
by the `conditional Chebishev inequality', hence relation (5.18) holds.
 
Lemma 5A follows from relation (5.17), part~a) of Lemma~6 and the
observation that the random vectors $\{\bar I_{n,k}^{(V,\e)}(f),\,
f\in\Cal F\}$, defined in $(5.9')$ have the same distribution for all
$V\subset\{1,\dots,k\}$ as the random vector $\{\bar I_{n,k}^{\e}(f),\,
f\in\Cal F\}$, considered in the formulation of Lemma~5A. Hence
$$
P\(\sup_{f\in\Cal F} |S(f)|>\frac A2n^k\sigma^{k+1}\)\le
2^kP\(\sup_{f\in\Cal F} \left|\bar I_{n,k}^{\e}(f)\right|
>2^{-(k+1)}A n^k\sigma^{k+1}\).
$$
\medskip
 
In the proof of Lemma~5B I apply the following Lemma~7B which is a
version of Lemma~7A.
\medskip\noindent
{\bf Lemma 7B.} {\it Let us consider a class of functions $\Cal F$
satisfying the conditions of Proposition~5 and the random variables
$\bar I_{n,k}^V(f,y)$, $f\in\Cal F$, $V\subset\{1,\dots,k\}$,
defined in formula~(5.3). Let $\Cal B=\Cal B(\xi_{1,s}^{(1)},
\dots, \xi_{n,s}^{(1)};\;1\le s\le k)$ denote the $\sigma$-algebra
generated by the random variables
$\xi_{1,s}^{(1)},\dots,\xi_{n,s}^{(1)}$, $1\le s\le k$, taking part
in the definition of the random variables $\bar
I_{n,k}^V(f,y)$ and $H_{n,k}^V(f)$.
\medskip
\item{a)} For all $V\subset\{1,\dots,k\}$, $V\neq\{1,\dots,k\}$,
there exists a number $A_0=A_0(k)>0$ such that the inequality
$$
P\(\sup_{f\in\Cal F} E(H^{V}_{n,k}(f)|\Cal B)
>2^{-(4k+4)}A^{(2k-1)/k} n^{2k}\sigma^{2k+2}\)<
n^{k-1}e^{-A^{1/2k} n\sigma^2/k}
\tag5.19
$$
holds for all $A\ge A_0$.
\medskip
\item{b)} Given two subsets $V_1,V_2\subset\{1,\dots,k\}$ of the
set $\{1,\dots,k\}$ define the random integrals
$$
H_{n,k}^{(V_1,V_2)}(f)=\int |\bar I_{n,k}^{V_1}(f,y)
\bar I_{n,k}^{V_2}(f,y)| \rho(\,dy),
\quad f\in\Cal F,
$$
with the help of the functions $\bar I_{n,k}^V(f,y)$ defined in
(5.3). If at least one of the sets $V_1$ and $V_2$ is not the
set $\{1,\dots,k\}$, then there exists some number $A_0=A_0(k)>0$
such that if the integrals $H_{n,k}(f)$, $f\in\Cal F$, determined by
this class of functions $\Cal F$ have a good tail behaviour at level
$T^{(2k+1)/2k}$ for some $T\ge A_0$, then the inequality
$$
P\(\sup_{f\in\Cal F} E(H^{(V_1,V_2)}_{n,k}(f)|\Cal B)
>2^{-(2k+2)}A^2n^{2k}\sigma^{2k+2}\)<2n^{k-1}e^{-A^{1/2k}n\sigma^2/k}
\tag5.20
$$
holds for all $A\ge T$.}
\medskip\noindent
{\it Proof of Lemma 7B.}\/ The proof of part~a) of Lemma 7B is similar
to that of Lemma~7A, only the formulas applied in it become a
little bit more complicated. Hence I omit it. (The difference
between the power of the parameter $A$ at the right-hand side of
formulas (5.19) and (5.12) appear, since the left-hand side of
(5.19) contains the term $A^{(2k-1)/2k}$ and not $A^2$.)
Part b) will be proved with the help of Part a) and the inequality
$$
\sup_{f\in\Cal F} E(H^{(V_1,V_2)}_{n,k}(f)|\Cal B) \le
\(\sup_{f\in\Cal F} E(H^{V_1}_{n,k}(f)|\Cal B)\)^{1/2}
\(\sup_{f\in\Cal F} E(H^{V_2}_{n,k}(f)|\Cal B)\)^{1/2}
$$
which follows from the Schwarz inequality applied for integrals with
respect to conditional distributions. Let us assume that
$V_1\neq\{1,\dots,k\}$. Then the last inequality implies that
$$
\aligned
&P\(\sup_{f\in\Cal F} E(H^{(V_1,V_2)}_{n,k}(f)|\Cal B)
>2^{-(2k+2)}A^2n^{2k}\sigma^{2k+2}\)\\
&\qquad \le P\(\sup_{f\in\Cal F} E(H^{V_1}_{n,k}(f)|\Cal B)
>2^{-(4k+4)}A^{(2k-1)/k} n^{2k}\sigma^{2k+2}\) \\
&\qquad\qquad+P\(\sup_{f\in\Cal F} E(H^{V_2}_{n,k}(f)|\Cal B)
>A^{(2k+1)/k} n^{2k}\sigma^{2k+2}\).
\endaligned
$$
Hence the estimate (5.19) for $V=V_1$ together with the inequality
$$
P\(\sup_{f\in\Cal F} E(H^{V_2}_{n,k}(f)|\Cal B)
>A^{(2k+1)/k} n^{2k}\sigma^{2k+2}\)\le n^{k-1} e^{-A^{1/2k}n\sigma^2/k}
$$
which follows from Part a) if $V_2\neq\{1,\dots,n\}$ (in this case
the level $A^{(2k+1)/k} n^{2k}\sigma^{2k+2}$ can be replaced by
$2^{-(4k+4)}A^{(2k-1)/k} n^{2k}\sigma^{2k+2}$ in the probability we
consider) and from the conditions of Part b) if
$V_2=\{1,\dots,k\}$ imply relation (5.20).
\medskip
Now I prove Lemma~5B.
\medskip\noindent
{\it Proof of Lemma 5B.}\/ By Part b) of Lemma~6 it is enough to
prove that relation (5.6) holds if the random variables $\bar W(f)$
are replaced in it by the random variables $W(f)$ defined in
formula~(5.11). We shall prove this by applying Lemma~3 with the
choice of $Z(f)=H_{n,k}^{(\bar V,\bar V)}(f)$, $\bar V=\{1,\dots,k\}$,
$\bar Z(f)=W(f)-Z(f)$, $f\in\Cal F$, $\Cal B=\Cal
B(\xi_{1,s}^{(1)},\dots, \xi_{n,s}^{(1)};\,1\le s\le k)$, and the set
$$
B=\bigcap\Sb (V_1,V_2)\colon V_j\subset \{1,\dots,k\},\;j=1,2\\
V_1\neq\{1,\dots,k\} \text { or } V_2\neq\{1,\dots,k\} \endSb
\left\{\oo\colon \sup_{f\in\Cal F} E(H^{(V_1,V_2)}_{n,k}(f)|\Cal B)(\oo)
\le 2^{-(2k+2)} A^{2} n^{2k}\sigma^{2k+2}\right\}.
$$
 
By Lemma 7B $1-P(B))\le2^{2k+1}n^{k-1} e^{-A^{1/2k}n\sigma^2/k}$,
and to prove Lemma 5B with the help of Lemma~3 it is enough to show
that
$$
P\(\left.|\bar Z(f)|>\frac{A^2}2 n^{2k}\sigma^{2(k+1)}\right|
\Cal B\)(\oo)\le\frac12 \quad \text{for all }f\in \Cal F \text { if }
\oo\in B.
$$
To prove this relation observe that
$$
E(|\bar Z(f)| |\Cal B)\le \summ \Sb (V_1,V_2)\colon V_j\subset
\{1,\dots,k\},\;j=1,2\\
V_1\neq\{1,\dots,k\} \text { or } V_2\neq\{1,\dots,k\} \endSb
E(H^{(V_1,V_2)}_{n,k}(f)|\Cal B)\le\frac{A^2}4n^{2k}\sigma^{2k+2}
\quad \text{if } \oo\in B
$$
for all $f\in \Cal F$. Hence the `conditional Markov inequality'
implies that
$$
P\(\left. |\bar Z(f)|> \frac{A^2}2n^{2k}\sigma^{2k+2}\right|\Cal B\)
\le\frac12 \quad\text{if }\oo\in B\quad \text{and } f\in\Cal F.
$$
Lemma~5B is proved.
 
\beginsection 6. The proof of Propositions 4 and 5
 
%\medskip\noindent
%{\bf 6. The proof of Propositions 4 and 5}
%\medskip\noindent
The proof of Propositions 4 and 5 for general $k\ge1$ with the help
of the symmetrization lemmas~5A and~5B is similar to the proof of
Proposition~4 in the case $k=1$ presented in Section~4. The proof
applies an induction procedure with respect to the parameter $k$.
In the proof of Proposition~4 for parameter~$k$ we may assume that
Propositions~4 and~5 hold for $k'<k$. In the proof of Proposition~5
we may also assume that Proposition~4 holds for the parameter~$k$.
 
In the proof of Proposition 4 let us introduce (with the notation of
this proposition) the functions
$$
S^2_{n,k}(f)(x_{j,s},\,1\le j\le n,\,1\le s\le k)=\frac1{k!}\summ\Sb
1\le j_s\le n,\; s=1,\dots, k\\ j_s\neq j_{s'} \text{ if } s\neq
s'\endSb f^2\(x_{j_1,1},\dots,x_{j_k,k}\),\quad f\in\Cal F, \tag6.1
$$
where $x_{j,s}\in X$, $1\le j\le n$, $1\le s\le k$. Fix some number
$A>T$, and define the set $H$
$$
\aligned
H=H(A)&=\biggl\{(x_{j,s},\,1\le j\le n,\,1\le s\le k), \\
&\qquad \sup_{f\in\Cal F} S^2_{n,k}(f)(x_{j,s},\,1\le j\le n,\,
1\le s\le k)>2^kA^{4/3}n^k\sigma^2\biggr\}.
\endaligned \tag6.2
$$
We want to show that
$$
P(\{\oo\colon (\xi_{j,s}(\oo),\,1\le j\le n,\,1\le s\le k)\in H\})\le
2^k e^{-A^{2/3k}n\sigma^2} \quad\text{if }A\ge T. \tag6.3
$$
 
Relation (6.3) will be proved by means of the Hoeffding decomposition
of the $U$-statistics with kernel functions $f^2(x_1,\dots,x_k)$,
$f\in\Cal F$, and by the estimation of the sum this decomposition
yields. More explicitly, write
$$
f^2(x_1,\dots,x_k)=\summ_{V\subset\{1,\dots,k\}} f_V(x_s,s\in V)
\tag6.4
$$
with
$$
f_V(x_s,s\in V)=\prodd_{s\notin V}P_{\mu,s}\prodd_{s\in V}
Q_{\mu,s}f^2(x_1,\dots,x_k), \tag6.5
$$
where $P_{\mu,s}$ and $Q_{\mu,s}$ are the operators $P_\mu$ and
$Q_\mu$ defined in formulas (2.7) and (2.8) if
$(Y_1\times Z\times Y_2,\Cal Y_1\times \Cal Z\times Y_2)$ is the
$k$-fold product $(X^k,\Cal X^k)$ of the measurable space $(X,\Cal X)$
in these definitions, $Y_1$ is the product of the first $s-1$
components, $Z$ is the $s$-th component and $Y_2$ is the product of
the last $k-s$ components in this product space. (Relation (6.4)
follows from the identity $f^2=\prodd_{s=1}^k(P_{\mu,s}+Q_{\mu,s})f^2$
if the multiplications
are carried out in this formula. In the calculation we exploit that
the operators $P_{\mu,s}$ and $P_{\mu,s'}$ are commutative if
$s\neq s'$, and the same relation holds for the pairs $P_{\mu,s}$
and $Q_{\mu,s'}$ or $Q_{\mu,s}$ and $P_{\mu,s'}$ or $Q_{\mu,s}$ and
$Q_{\mu,s'}$.)
 
The identity $S^2_{n,k}(f)(\xi_{j,r}\,1\le j\le n,1\le r\le
k)=k!I_{n,k}(f^2)$ holds for all $f\in\Cal F$, and by writing the
(Hoeffding type) decomposition (6.4) for each term
$f^2(\xi_{j_1,1}\dots,\xi_{j_k,k})$ in the expression
$I_{n,k}(f^2)$ we get that
$$
\aligned
&P\(\sup_{f\in\Cal F}S^2_{n,k}(f)(\xi_{j,s},\,1\le j\le n,\,1\le s\le k)
>2^kA^{4/3}n^k\sigma^2\)\\
&\qquad \le\summ_{V\subset\{1,\dots,k\}} P\(\sup_{f\in\Cal F}
n^{k-|V|}|\bar I_{n,|V|}(f_V)|>A^{4/3}n^k\sigma^2\)
\endaligned \tag6.6
$$
with the functions $f_V$ defined in (6.5). We want to give a good
estimate for all terms in the sum at the right-hand side in (6.6).
For this goal we show that the classes of functions $\Cal
F_V=\{f_V\colon f\in \Cal F\}$ satisfy the conditions of Proposition~4
for all $V\subset\{1,\dots,k\}$.
 
The functions $f_V$ are canonical for all $V\subset\{1,\dots,k\}$.
(This follows from the commutativity relations between the
operators $P_{\mu,j}$ and $Q_{\mu,j}$ mentioned before, the identity
$P_{\mu,j}Q_{\mu,j}=0$
and the fact that the canonical property of the function can be
expressed in the form $P_{\mu,j}f_V=0$ for all $j\in V$.) We have
$|f^2(x_1,\dots,x_k)|\le 2^{-2(k+1)}$. The norm of $Q_{\mu,j}$ as a
map from the $L_\infty$ space to $L_\infty$ space is less than 2,
the norm of $P_{\mu,j}$ is less than 1, hence $\left|\supp_{x_j\in
X,j\in V}f_V(x_j,j\in V)\right|\le 2^{-(k+2)}\le2^{-(k+1)}$ for all
$V\subset\{1,\dots,k\}$. We have $\int
f^4(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)\le 2^{-(k+1)}\sigma^2$,
hence $\int f^2_V(x_j,j\in V)\prodd_{j\in V}\mu(\,dx_j)\le
2^{-(k+1)}\sigma^2\le\sigma^2$ for all $V\subset\{1,\dots,k\}$ by
Lemma~1. Finally, to check that the class of functions $\Cal
F_V=\{f_V\colon f\in\Cal F\}$ is $L_2$-dense with exponent $L$ and
parameter $D$ observe that for all probability measures $\rho$ on
$(X^k,\Cal X^k)$ and pairs of functions $f,g\in \Cal F$
$\int(f^2-g^2)^2\,d\rho\le 2^{-2k}\int(f-g)^2\,d\rho$. This implies
that if $\{f_1,\dots,f_m\}$, $m\le D\e^{-L}$, is an $\e$-dense
subset of $\Cal F$ in the space $L_2(X^k,\Cal X^k,\rho)$, then the
set of functions $\{2^kf_1^2,\dots,2^kf_m^2\}$ is an $\e$-dense
subset of the class of functions $\Cal F'=\{2^kf^2\colon f\in \Cal F\}$
in the same space. Then by Lemma~1 and formula~(6.5) the set of
functions
$\{(f_1)_V,\dots,(f_m)_V)$ is an $\e$-dense subset of the class of
functions $\Cal F_V$ in the space $L_2(X^k,\Cal X^k,\rho)$
for all $V\subset\{1,\dots,k\}$. This means that $\Cal F_V$ is also
$L_2$-dense with exponent $L$ and parameter~$D$.
 
For $V=\emptyset$ the relation $f_V=\int f^2(x_1,\dots,x_k)
\mu(\,dx_1)\dots\mu(\,dx_k)\le\sigma^2$ holds, and
$I_{|V|}(f_{|V|})|=f_V\le\sigma^2$. Therefore the term corresponding
to $V=\emptyset$ in the sum at the right-hand side
of (6.6) equals zero if $A_0\ge1$ in the conditions of Proposition~4.
The terms corresponding to sets $V$, $1\le|V|\le k$, in these sums
satisfy the inequality
$$
\align
&P\(\sup_{f\in\Cal F}|\bar I_{n,|V|}(f_V)|>A^{4/3}n^{|V|}\sigma^2\)\\
&\qquad \le P\(\sup_{f\in\Cal F}
|\bar I_{n,|V|}(f_V)|>A^{4/3}n^{|V|}\sigma^{|V|+1}\)
\le e^{-A^{2/3k}n\sigma^2} \quad\text{if } 1\le|V|\le k.
\endalign
$$
This inequality follows from the inductive hypothesis if $|V|<k$,
and in the case $V=\{1,\dots,k\}$  from the inequality $A\ge
T$ and the assumption that $U$-statistics determined by a class of
functions satisfying the conditions of Proposition~4 have a good
tail behaviour at level $T^{4/3}$. The last relation together with
formula (6.6) imply relation~(6.3).
 
By conditioning the probability $P\(\left|\bar I_{n,k}^{\e}(f)
\right|>2^{-(k+2)}An^{k/2}\sigma^{k+1}\)$ with respect to the
random variables $\xi_{j,s}$, $1\le j\le n$, $1\le s\le k$, we get
with the help of Lemma~2 that
$$
\align
&P\(\left.\left|\bar I_{n,k}^{\e}(f)\right|
>2^{-(k+2)}A n^k\sigma^{k+1}\right|\xi_{j,s}(\oo)=x_{j,s},
1\le j\le n,1\le s\le k\) \\
&\qquad \le C\exp\left\{-B\(\frac{A^2n^{2k}\sigma^{2(k+1)}}{2^{2k+4}
S^2_{n,k}(x_{j,s},1\le j\le n,1\le s\le k)}\)^{1/k}\right\} \tag6.7 \\
&\qquad \le Ce^{-2^{-3-4/k}BA^{2/3k}n\sigma^2} \quad \text{for all }
f\in\Cal F\quad \text{if }\{x_{j,s},\, 1\le j\le n,\,1\le s\le
k\}\notin H.
\endalign
$$
Given some points $x_{j,s}$, $1\le j\le n$, $1\le s\le k$, define the
probability measures $\rho_s$, $1\le s\le k$, uniformly distributed
on the set $x_{j,s}$, $1\le j\le s$, i.e. $\rho_s(x_{j,s})=\frac1n$,
$1\le j\le n$, and their product $\rho=\rho_1\times\cdots\times\rho_k$.
If $f$ is a function on $(X^k,\Cal X^k)$ such that $\int f^2
\,d\rho\le\delta^2$ with some $\delta>0$, then $|f(x_{j,s})|
\le \delta n^{k/2}$ for all $1\le s\le k$, $1\le j\le n$, and
$P\(\left.\left|\bar I_{n,k}^{\e}(f)\right|>\delta n^{3k/2}
\right|\xi_{j,s}=x_{j,s}, 1\le j\le n,\, 1\le s\le k\)=0$. Choose
the numbers $\bar\delta=An^{-k/2}2^{-(k+2)}\sigma^{k+1}$ and
$\delta=2^{-(k+2)}n^{-k-1/2}\le\bar\delta$. (The inequality
$\delta\le\bar\delta$ holds, since $A\ge A_0\ge1$, and $\sigma\ge
n^{-1/2}$.)  Choose a $\delta$-dense set $\{f_1,\dots,f_m\}$ in the
$L_2(X^k,\Cal X^k,\rho)$ space with $m\le D\delta^{-L}\le 2^{(k+2)L}
n^{\beta+(k+1/2)L}$ elements. Then formula~(6.7) implies that
$$
\align
&P\(\sup_{f\in\Cal F}\left.\left|\bar I_{n,k}^{\e}(f)\right|
>2^{-(k+1)}A n^k\sigma^{k+1}\right|\xi_{j,s}(\oo)=x_{j,s},
1\le j\le n,1\le s\le k\) \\
&\quad \le \sum_{j=1}^m P\(\left.\left|\bar I_{n,k}^{\e}(f_j)\right|
>2^{-(k+2)}A n^k\sigma^{k+1}\right|\xi_{j,s}(\oo)=x_{j,s},
1\le j\le n,1\le s\le k\)  \tag6.8    \\
&\qquad \le C 2^{(k+2)L}n^{\beta+(k+1/2)L}
e^{-2^{-3-4/k}BA^{2/3k}n\sigma^2} \quad \text{if }\{x_{j,s},\, 1\le
j\le n,\,1\le s\le k\}\notin H.
\endalign
$$
 
Relations (6.3) and (6.8) imply that
$$
\aligned
&P\(\sup_{f\in\Cal F}\left|\bar I_{n,k}^{\e}(f)\right|
>2^{-(k+1)}A n^k\sigma^{k+1}\) \\
&\qquad \le C2^{(k+2)L}n^{\beta+(k+1/2)L}
e^{-2^{-3-4/k}BA^{2/3k}n\sigma^2}+
2^k e^{-A^{2/3k}n\sigma^2} \quad\text{if }A\ge T.
\endaligned \tag6.9
$$
Proposition 4 follows from the estimates (5.2) and (6.9) if the
constants $A_0$ and $K$ in the condition $n\sigma^2\ge K(L+\beta)
\log n$ are chosen sufficiently large. In this case the upper
bound these estimates yield for the probability at the left-hand side
of (3.11) is smaller than $e^{-A^{2/k}n\sigma^2}$.
\medskip
 
Let us turn to the proof of Proposition~5. By formula (5.8) it is
enough to show that
$$
\aligned
&P\(\sup_{f\in\Cal F} \left |H_{n,k}(f|G,V_1,V_2)\right|
>\frac{A^2}{2^{4k+1}k!} n^{2k}\sigma^{2(k+1)}\) \le
e^{-A^{1/2k}n\sigma^2}\\
&\qquad\text{ for all } G\in \Cal G\quad \text{and }
\;V_1,V_2\in\{1,\dots,k\} \quad\text{if } A\ge A_0
\endaligned \tag6.10
$$
with the random variables $H_{n,k}(f|G,V_1,V_2)$ defined in formula
(5.7). Let us first prove (6.10) in the case when $|e(G)|=k$, i.e.\ if
all vertices of the diagram $G$ are an end-point of some edge, and
the expression $H_{n,k}(f|G,V_1,V_2)$ contains no `symmetryzing term'
$\e_j$.  By the Schwarz inequality
$$
\aligned
|H_{n,k}(f|G,V_1,V_2)|&\le
\(\sum\Sb j_1,\dots,j_k, 1\le j_s\le n,\\ j_s\neq j_{s'} \text{ if
} s\neq s'\endSb \int
f^2(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y)
\rho(\,dy)\)^{1/2}\\
&\qquad \(\sum\Sb j_1,\dots,j_k, 1\le j_s\le n,\\ j_s\neq j_{s'}
\text{ if }s\neq s'\endSb \int f^2(\xi_{j_1,1}^{(\bar\delta_1)},
\dots,\xi_{j_k,k}^{(\bar\delta_k)},y) \rho(\,dy)\)^{1/2},
\endaligned \tag6.11
$$
for such diagrams $G$, where $\delta_s=1$ if $s\in V_1$,
$\delta_s=-1$ if $s\notin V_1$, and $\bar\delta_s=1$ if $s\in V_2$,
$\bar\delta_s=-1$ if $s\notin V_2$. Hence
$$
\align
&\left\{\oo\colon \sup_{f\in\Cal F} \left
|H_{n,k}(f|G,V_1,V_2)(\oo)\right|
>\frac{A^2}{2^{4k+1}k!} n^{2k}\sigma^{2(k+1)}\right\} \\
&\quad \subset
\left\{\oo\colon\sup_{f\in\Cal F} \sum\Sb j_1,\dots,j_k, 1\le j_s\le
n,\\ j_s\neq j_{s'} \text{ if } s\neq s'\endSb \int
f^2(\xi_{j_1,1}^{(\delta_1)}(\oo),\dots,\xi_{j_k,k}^{(\delta_k)}
(\oo),y) \rho(\,dy)>\frac {A^2n^{2k}\sigma^{2(k+1)}}
{2^{4k+1}k!} \right\}\\
&\qquad \cup
\left\{\oo\colon\sup_{f\in\Cal F} \sum\Sb j_1,\dots,j_k, 1\le j_s\le
n,\\ j_s\neq j_{s'} \text{ if } s\neq s'\endSb \int
f^2(\xi_{j_1,1}^{(\bar\delta_1)}(\oo),\dots,\xi_{j_k,k}^{(\bar\delta_k)}
(\oo),y)
\rho(\,dy)>\frac{A^2n^{2k}\sigma^{2(k+1)}}{2^{4k+1}k!}\right\}.
\endalign
$$
 
The last relation implies that
$$
\align
&P\(\sup_{f\in\Cal F} \left |H_{n,k}(f|G,V_1,V_2)\right|
>\frac{A^2}{2^{4k+1}k!} n^{2k}\sigma^{2(k+1)}\) \\
&\qquad \le 2P\(\sup_{f\in\Cal F} \sum\Sb j_1,\dots,j_k,
1\le j_s\le n,\\ j_s\neq j_{s'} \text{ if } s\neq s'\endSb
h_f(\xi_{j_1,1},\dots,\xi_{j_k,k})
>\frac{A^2n^{2k}\sigma^{2(k+1)}}{2^{4k+1}k!}\) \tag6.12
\endalign
$$
with the function $h_f(x_1,\dots,x_k)=\int
f^2(x_1,\dots,x_k,y)\rho(\,dy)$,
$f\in\Cal F$. (In this upper bound we could get rid of the
terms $\delta_j$ and $\bar\delta_j$, i.e. on the dependence of the
expression $H_{n,k}(f|G,V_1,V_2)$ on the sets $V_1$ and $V_2$, since
the probability of the events in the previous formula do not depend
on these terms.)
 
I claim that
$$
P\(\supp_{f\in\Cal F} |\bar I_{n,k}(h_f)|\ge An^k \sigma^2\)\le
2^k e^{-A^{1/2k}n\sigma^2} \quad \text{for }A\ge A_0  \tag6.13
$$
if the constant $A_0$ and $K$ are chosen sufficiently large in
Proposition~5. Relation (6.13) together with the relation
$\frac{n^{2k}\sigma^{2(k+1)}}{2^{4k+1}k!}\ge n^k\sigma^2$ imply
that the probability at the right-hand side of (6.12) can be
bounded by $2^{k+1}e^{-A^{1/k}n\sigma^2}$, and the estimate
(6.10) holds in the case $|e(G)|=k$. Relation (6.13)
can be proved similarly to formula (6.3) in the proof of
Proposition~4. It is not difficult to check that $0\le\int
h_f(x_1,\dots,x_k) \mu(\,dx_1)\dots\mu(\,dx_k)\le\sigma^2$,
$\sup|h_f(x_1,\dots,x_k)|\le 2^{-2(k+1)}$, and the class of functions
$\Cal H=\{2^kh_f,\; f\in\Cal F\}$ is an $L_2$-dense class with
exponent $L$ and parameter $D$. This means that by applying the
(Hoeffding type) decomposition of the functions $h_f$, $f\in \Cal F$,
similarly to formula (6.4) we get such sets of functions $(h_f)_V$,
$f\in\Cal F$ for all $V\subset \{1,\dots,k\}$ which satisfy
the conditions of Proposition~4. Hence a natural adaptation of the
estimate given for the expression at the right-hand side of (6.6)
yields the proof of formula (6.13). Let us observe that by our
inductive hypothesis the result of Proposition~4 holds also for $k$,
and this allows us to carry out the estimates we need
also for the class of functions $(h_f)_V$, $f\in\Cal F$, with
$V=\{1,\dots,k\}$ if $A\ge A_0$.
 
In the case $e(G)<k$ formula (6.10) will be proved with the help of
Lemma~2. To carry out this proof first an appropriate expression
will be introduced and bounded for all sets $V_1,V_2\subset
\{1,\dots,k\}$ and diagrams $G\in \Cal G$ such that $|e(G)|<k$. To
define the  expression we shall bound first some notations will be
introduced.
 
Let us consider the set $J_0(G)=J_0(G,k,n)$,
$$
\align
J_0(G)&=\{(j_1,\dots,j_k,j'_1,\dots,j'_k)\colon1\le j_s,j'_s\le n,\,
1\le s\le k,\, j_s\neq j_{s'}\text { if } s\neq s', \\
&\qquad j'_s\neq j'_{s'}\text{ if }s\neq s',\, j_s=j'_{s'}\text{ if }
(s,s')\in G,\; j_s\neq j'_{s'}\text{ if } (s,s')\notin G\},
\endalign
$$
the set of those sequences $(j_1,\dots,j_k,j'_1,\dots,j'_k)$
which appear as indices in the summation in formula (5.7). I introduce
a partition of $J_0(G)$ appropriate for our purposes.
 
To do this first I define the sets
$M_1=M_1(G)=\{s(1),\dots,s(k-|e(G)|)\}=\{1,\dots,k\}\setminus
v_1(G)$, $s(1)<\cdots<s(k-|e(G)|)$, and
$M_2=M_2(G)=\{\bar s(1),\dots,\bar s(k-|e(G|)\}=\{1,\dots,k\}\setminus
v_2(G)$, $\bar s(1)<\cdots<\bar s(k-|e(G|)$, the sets of those vertices
of the first and second row of the diagram $G$ in increasing order
from which no edges start. Let us also introduce the set
$V(G)=V(G,n,k)$,
$$
\align
V(G)&=\{(j_{s(1)},\dots,j_{s(k-|e(G)|)},
j'_{\bar s(1)},\dots,j'_{\bar s(k-|e(G)|)})\colon1\le j_{s(p)},
j'_{\bar s(p)}\le n,\\
&\quad 1\le p\le k-|e(G)|,\, j_{s(p)}\neq j_{s(p')},\,
j'_{\bar s(p)}\neq j'_{\bar s(p')} \text { if }p\neq p',\,
1\le p,p'\le k-|e(G)|\\
&\qquad j_{s(p)}\neq j'_{\bar s(p')},\;1\le p,p'\le k-|e(G)|\},
\endalign
$$
which is the set consisting from the restriction of the coordinates of
the vectors
$$
(j_1,\dots,,j_k,j'_1,\dots,j'_k)\in J_0(G)
$$
to $M_1\cup M_2$. Given a vector $v\in V(G)$ let $v(r)$,
$1\le r\le k-|e(G)|$, and $\bar v(r)$, $1\le r\le k-|e(G)|$, denote
its coordinates corresponding to the set $M_1$ and $M_2$ respectively.
Put
$$
\align
E_G(v)&=\{(j_1,\dots,j_k,j'_1,\dots,j'_k)\colon1\le j_s\le n, \; 1\le
j'_{s'}\le n,\quad 1\le s,s'\le k,\\
&\qquad j_s\neq j_{s'}\text{ if }s\neq s',\; j'_{\bar s}\neq
j'_{\bar s'}\text{ if }\bar s\neq\bar  s', \\
&\qquad j_s=j'_{s'}\text{ if } (s,s')\in e(G)\text { and } j_s\neq
j'_{s'}\text{ if } (s,s')\notin e(G)\\
&\qquad j_{s(r)}=v(r),\, j'_{\bar s(r)}=\bar v(r),\, 1\le r\le
k-|e(G)|\},\quad v\in V(G),
\endalign
$$
where $\{s(1),\dots,s(k-|e(G)|)\}=M_1$, $\{\bar s(1),\dots,
\bar s(k-|e(G)|)\}=M_2$ in the last line of this definition.  The set
$E_G(v)$ contains those vectors in $J_0(G)$ whose coordinates in
$M_1\cup M_2$ are prescribed by the vector $v\in V(G)$, the
remaining coordinates can be put into pairs $(s,{s'})$ such that
$(s,s')\in e(G)$, and the values $j_s=j'_{s'}$, $(s,s')\in e(G)$,
must be different for different pairs, otherwise they can be chosen
freely from the set $\{1,\dots,n\}\setminus\{v(1),\dots,v(k-|e(G)|),
\bar v(1),\dots,\bar v(k-|e(G)|)\}$.
 
Now we define the partition
$$
J_0(G)=\bigcup_{v\in V(G)} E_G(v)
$$
of the set $J_0(G)$.
 
The inequality
$$
P\(S(\Cal F|G,V_1,V_2))>A^{8/3}n^{2k}\sigma^4\)\le
2^{k+1}e^{-A^{2/3k}n\sigma^2} \quad\text{if }A\ge A_0\text{ and }
e(G)<k \tag6.14
$$
will be proved for the random variable
$$
\align
S(\Cal F|G,V_1,V_2)=\sup_{f\in\Cal F}\sum_{v\in V(G)}
\biggl(&\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in E_G(v)}
\int f(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y) \\
&\qquad f(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}
^{(\bar\delta_k)},y) \rho(\,dy)\biggr)^2,
\endalign
$$
where $\delta_s=1$ if $s\in V_1$, $\delta_s=-1$ if $s\notin V_1$,
and $\bar\delta_s=1$ if $s\in V_2$, $\bar\delta_s=-1$ if $s\notin V_2$.
 
 
To prove formula (6.14) let us first fix some $v\in V(G)$ and apply
the Schwarz inequality. It yields that
$$
\align
&\(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in E_G(v)}
\int f(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y)
f(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}
^{(\bar\delta_k)},y) \rho(\,dy)\)^2\\
&\qquad\le
\(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in E_G(v)}
\int f^2(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y)
\rho(\,dy)\) \\
&\qquad\qquad \(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in E_G(v)}
\int f^2(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}
^{(\bar\delta_k)},y) \rho(\,dy)\).
\endalign
$$
for all $v\in V(G)$. Summing up these inequalities for all
$v\in V(G)$ we get that
$$
\aligned
S(\Cal F|G,V_1,V_2)&\le\sup_{f\in\Cal F}\sum_{v\in V(G)}
\(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in E_G(v)}
\int f^2(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y)
\rho(\,dy)\) \\
&\qquad\(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in E_G(v)}
\int f^2(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}
^{(\bar\delta_k)},y) \rho(\,dy)\)    \\
&\le \supp_{f\in\Cal F} \(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in
J_0(G)} \int f^2(\xi_{j_1,1}^{(\delta_1)},\dots,
\xi_{j_k,k}^{(\delta_k)},y) \rho(\,dy)\) \\
&\qquad \qquad \supp_{f\in\Cal F}
\(\sum_{(j_1,\dots,j_k,j'_1,\dots,j'_k)\in J_0(G)}
\int f^2(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}
^{(\bar\delta_k)},y) \rho(\,dy)\)
\endaligned \tag6.15
$$
To check formula (6.15) we have to observe that by multiplying the
inner sum at the left-hand side of this inequality each term
$f^2(\xi_{j_1,1}^{(\delta_1)},\dots,\xi_{j_k,k}^{(\delta_k)},y)
f^2(\xi_{j'_1,1}^{(\bar\delta_1)},\dots,\xi_{j'_k,k}
^{(\bar\delta_k)},y)$ appears only once. (In particular, it is
determined which index $v\in V(G)$ has to be taken in the outer sum
to get this term. The coordinates of this vector $v$ agree with the
coordinates of the vector $j=(j_1,\dots,j_k,j'_1,\dots,j'_k)$ in
$M_1\cup M_2$, with the coordinates of the vector $j$ which
correspond to those vertices from which no edges of the diagram $G$
start.) Beside this, all these products appear if the multiplications
at the right-hand side expression are carried out.
 
Relation (6.15) implies that
$$
P(S(\Cal F|G,V_1,V_2)>A^{8/3}n^{2k}\sigma^4) \le
2P\(\supp_{f\in\Cal F} \bar I_{n,k}(h_f)>A^{4/3}n^k\sigma^2\)
$$
with $h_f(x_1,\dots,x_k)=\int f^2(x_1,\dots,x_k,y)\rho(\,dy)$.
(Here we exploited that in the last formula $S(\Cal F|G,V_1,V_2)$
is bounded by the product of two random variables whose distributions
do not depend on the sets $V_1$ and $V_2$.) Thus to prove inequality
(6.14) it is enough to show that
$$
2P\(\supp_{f\in\Cal F} \bar I_{n,k}(h_f)>A^{4/3}n^k\sigma^2\)\le
2^{k+1}e^{-A^{2/3k}} \quad \text{if } A\ge A_0. \tag6.16
$$
Actually formula (6.16) has been already proved, only formula (6.13)
has to be applied, and  the parameter $A$ has to be replaced by
$A^{4/3}$ in it.
 
The proof of Proposition~5 can be completed similarly to
Proposition~4. It follows from Lemma~2 that
$$
\aligned
&P\(\left.|H_{n,k}(f|G,V_1,V_2)|
>\frac{A^2}{2^{4k+2}k!} n^{2k}\sigma^{2(k+1)}
\right| \xi^{\pm1}_{j,s},\,1\le j\le n,\,1\le s\le k\)(\oo)\\
&\qquad \le Ce^{-B2^{-(4+2/k)}(k!)^{-1/k} A^{2/3k}n\sigma^2} \quad
\text{if}\quad S(\Cal F|G,V_1,V_2)(\oo)\le A^{8/3}n^{2k}\sigma^4 \\
&\qquad\qquad\text{ for all } f\in\Cal F,\; G\in \Cal G,\;
|e(G)|<k, \quad \text{and } \;V_1,V_2\in\{1,\dots,k\}
\quad\text{if } A\ge A_0.
\endaligned \tag6.17
$$
Indeed, in this case the conditional probability considered in (6.17)
can be bounded by $C\exp\left\{-B\(\frac{A^4n^{4k}\sigma^{4(k+1)}}
{2^{8k+4}(k!)^2S(\Cal F|G,V_1,V_2)}\)^{1/2j}\right\}\le C\exp
\left\{-B\(\frac{A^{4/3}n^{2k}\sigma^{4k}}{2^{8k+4}(k!)^2}\)^{1/2j}
\right\}$, where $2j=2k-2|e(G)|$, the number of vertices of the
diagram $G$ from which no edges start. Since $j\le k$, $n\sigma^2\ge1$,
and also $\frac{A^{4/3}}{2^{8k+4}(k!)^2}\ge1$ if $A_0$ is chosen
sufficiently large the above calculation implies relation~(6.17).
 
Let us show that also the inequality
$$
\aligned
&P\(\left.\sup_{f\in\Cal F} |H_{n,k}(f|G,V_1,V_2)|
>\frac{A^2}{2^{4k+1}k!} n^{2k}\sigma^{2(k+1)}
\right| \xi^{\pm1}_{j,s},\,1\le j\le n,\,1\le s\le k\)(\oo)\\
&\qquad \le Cn^{(3k+1)L/2+\beta}
e^{-BA^{2/3k}n\sigma^2/2^{(4+2/k)}(k!)^{1/k}} \quad
\text{if } S(\Cal F|G,V_1,V_2))(\oo)\le A^{8/3}n^{2k}\sigma^4 \\
&\qquad \qquad\text{ for all } G\in \Cal G,\; |e(G)|<1, \quad
\text{and } \;V_1,V_2\in\{1,\dots,k\} \quad\text{if } A\ge A_0
\endaligned \tag6.18
$$
holds.
 
To prove formula (6.18) let us fix an elementary event
$\oo\in\Omega$ which satisfies the relation
$S(\Cal F|G,V_1,V_2))(\oo)\le A^{8/3}n^{2k}\sigma^4$,
two sets $V_1,V_2\subset\{1,\dots,k\}$, a
diagram $G$, consider the points $x_{j,s}^{(\pm1)}=
x_{j,s}^{(\pm1)}(\oo)=\xi_{j,s}^{(\pm1)}(\oo)$,
$1\le j\le n$, $1\le s\le k$, and introduce with their help the
following probability measures: For all $1\le s\le k$ define the
probability measures $\nu_s^{(1)}$ which are
uniformly distributed on the points $x_{j,s}^{(\delta_s)}$, $1\le j\le
n$, and $\nu_s^{(2)}$ which are uniformly distributed
on the points $x_{j,s}^{(\bar\delta_s)}$, $1\le j\le n$, where
$\delta_s=1$ if $s\in V_1$, $\delta_s=-1$ if $s\notin V_1$, and
similarly $\bar\delta_s=1$ if $s\in V_2$ and $\bar\delta_s=-1$ if
$s\notin V_2$. Let us consider the product measures
$\alpha_1=\nu_1^{(1)}\times\cdots\times\nu_k^{(1)}\times\rho$,
$\alpha_2=\nu_1^{(2)}\times\cdots\times\nu_k^{(2)}\times\rho$ on
the product space $(X^k\times Y,\Cal X^k\times\Cal Y)$, where $\rho$
is that probability measure on $(Y,\Cal Y)$ which appears in
Proposition~5, together with the measure
$\alpha=\frac{\alpha_1+\alpha_2}2$. Given two functions $f\in \Cal F$
and $g\in\Cal F$ we  give an upper bound for
$|H_{n,k}(f|G,V_1,V_2)(\oo)-H_{n,k}(g|G,V_1,V_2)(\oo)|$ if $\int
(f-g)^2\,d\alpha\le\delta$ with some $\delta>0$. (This bound
does not depend on the `randomizing terms' $\e_j(\oo)$ in the
definition of the random variable $H_{n,k}(\cdot|G,V_1,V_2)$.)
 
In this case $\int(f-g)^2\,d\alpha_j\le2\delta^2$, and
$$
\align
\int&\left|f(x_{1,j_1}^{(\delta_1)},\dots,x_{k,j_k}^{(\delta_k)},y)-
g(x_{1,j_1}^{(\delta_1)},\dots,x_{k,j_k}^{(\delta_k)},y)\right|^2
\rho(\,dy) \le2\delta^2n^k, \\
\int& \left|f(x_{1,j_1}^{(\delta_1)},\dots,x_{k,j_k}^{(\delta_k)},y)-
g(x_{1,j_1}^{(\delta_1)},\dots,x_{k,j_k}^{(\delta_k)},y)\right|
\rho(\,dy) \le\sqrt2\delta n^{k/2}
\endalign
$$
for all $1\le s\le k$, and $1\le j_s\le n$, and the same result
holds if all $\delta_s$ is replaced by $\bar\delta_s$, $1\le s\le
k$. Since $|f|\le1$ for $f\in\Cal F$, the condition
$\int(f-g)^2\,d\alpha\le \delta^2$ implies that
$$
\align
\int &\biggl|f(\xi_{j_1,1}^{(\delta_1)}(\oo),\dots,
\xi_{j_k,k}^{(\delta_k)}(\oo),y)
f(\xi_{j'_1,1}^{(\bar\delta_1)}(\oo),\dots,
\xi_{j'_k,k}^{(\bar\delta_k)}(\oo),y) )\\
&\qquad -g(\xi_{j_1,1}^{(\delta_1)}(\oo),\dots,
\xi_{j_k,k}^{(\delta_k)}(\oo),y)
g(\xi_{j'_1,1}^{(\bar\delta_1)}(\oo),\dots,
\xi_{j'_k,k}^{(\bar\delta_k)}(\oo),y)\biggr| \rho(\,dy)
\le2\sqrt2\delta n^{k/2}
\endalign
$$
for all vectors $(j_1,\dots,j_k,j'_1,\dots,j'_k)$ which appear as an
index in the summation in (5.7), and
$$
|H_{n,k}(f|G,V_1,V_2)(\oo)-H_{n,k}(g|G,V_1,V_2)(\oo)|
\le2\sqrt2\delta n^{5k/2}
$$
if the originally fixed $\oo\in\Omega$ is considered.
 
Put $\bar\delta=\frac{A^2 n^{-k/2}\sigma^{2(k+1)}}{2^{(4k+7/2)} k!}$,
and $\delta=n^{-(3k+1)/2}\le\bar\delta$ (since $\sigma\ge
\frac1{\sqrt n}$ and we may assume that $A\ge A_0$ is sufficiently
large), choose a $\delta$-dense subset $\{f_1,\dots,f_m\}$ in the
$L_2(X^k\times Y,\Cal X^k\times Y,\alpha)$ space with $m\le
D\delta^{-L}\le n^{(3k+1)L/2+\beta}$ elements. Relation (6.17) for
these functions together with the above estimates yield formula (6.18).
 
It follows from relations (6.14) and (6.18) that
$$
\align
&P\(\sup_{f\in\Cal F}|H_{n,k}(f|G,V_1,V_2)|
>\frac{A^2}{2^{4k+1}k!} n^{2k}\sigma^{2(k+1)}\)\le
2^{k+1}e^{-A^{2/3k}n\sigma^2}\\
&\qquad + Cn^{(3k+1)L/2+\beta}
e^{-BA^{2/3k}n\sigma^2/2^{(4+2/k)}(k!)^{1/k}}
\quad\text{if }A\ge A_0
\endalign
$$
for all $V_1,V_2\subset\{1,\dots,k\}$ also in the case $|e(G)|\le k-1$.
This means that relation (6.10) holds also in this case if the constants
$A_0$ and $K$ are chosen sufficiently large in Proposition~5.
Proposition~5 is proved.
 
\beginsection Appendix. \ The proof of Proposition 3
 
I shall explain the proof of Proposition 3 in a concise form. A
more detailed explanation can be found in~[8].
 
\medskip\noindent
{\it The proof of Proposition 3.}\/ Let us first introduce the
(random) probability measures $\delta_{\xi_j}$, $1\le j\le n$,
concentrated in the sample points $\xi_j$. We can write
$\mu_n-\mu=\frac1n\(\summ_{j=1}^n\(\delta_{\xi_j}-\mu\)\)$, and
formula (1.2) can be rewritten as
$$
\aligned
J_{n,k}(f)=\dfrac1{n^{k/2}k!}&\sum\Sb (j_1,\dots,j_k)\\ 1\le j_s\le n
\text{ for all }1\le s\le k\endSb \int' f(x_1,\dots,x_k) \\
&\qquad \(\delta_{\xi_{j_1}}(\,dx_1)-\mu(\,dx_1)\)\dots
\(\delta_{\xi_{j_k}}(\,dx_k)-\mu(\,dx_k)\).
\endaligned  \tag A1
$$
To rearrange the above sum in a way more appropriate for us let us
introduce the following notations: Let $\Cal P=\Cal P_k$ denote
the set of all partitions of the set $\{1,2,\dots,k\}$, and given
a sequence $(j_1,\dots,j_k)$, $1\le j_s\le n$, $1\le s\le k$, of
length $k$ let $H(j_1,\dots,j_k)$ denote that partition of
$\Cal P_k$ in which two points $s$ and $t$, $1\le s,t\le k$, belong
the same element of the partition if and only if $j_s=j_t$. Given a
set $A$, let $|A|$ denote its cardinality.
 
Let us rewrite the expression (A1) for $J_{n,k}(f)$ in the form
$$
\align
J_{n,k}(f)=\dfrac1{n^{k/2}k!}\sum_{P\in \Cal P} &\sum\Sb (j_1,\dots,
j_k),\\ 1\le j_s\le n,\, 1\le s\le k \\ H(j_1,\dots,j_k)=P \endSb
\int' f(x_1,\dots,x_k)   \tag  A2 \\
&\qquad \(\delta_{\xi_{j_1}}(\,dx_1)-\mu(\,dx_1)\)\dots
\(\delta_{\xi_{j_k}}(\,dx_k)-\mu(\,dx_k)\).
\endalign
$$
 
Let us remember that the diagonals $x_s=x_t$, $s\neq t$, were
omitted from the domain of integration in the formula defining
$J_{n,k}(f)$. This implies that in the case $j_s=j_t$ the measure
$\delta_{\xi_{j_s}}(\,dx_s)\delta_{\xi_{j_t}}(\,dx_t)$ has zero measure
in the domain of integration. We have to understand the cancellation
effects caused by this relation. I want to show that because of
these cancellations the expression in formula (A2) can be rewritten
as a linear combination of the degenerate $U$-statistics
$I_{n,|V|}(f_V)$ defined in (2.11) with not too large coefficients.
This seems to be a natural approach, but the detailed proof demands
some rather unpleasant calculations.
 
Let us fix some $P\in\Cal P$ and investigate the inner sum at the
right-hand side of~(A2) corresponding to this partition~$P$. For
the sake of simplicity let us first consider such an inner sum that
corresponds to a partition $P\in\Cal P$ which contains a set of the
form $\{1,\dots,s\}$ with some $s\ge2$. The products of measures
corresponding to the terms in the sum determined by such a
partition contain a part of length $s$ which has the form
$\(\delta_{\xi_{j}}(dx_1)-\mu(dx_1)\)\dots
\(\delta_{\xi_{j}}(dx_s)-\mu(dx_s)\)$
with some $1\le j\le n$. This part of the product can be rewritten
in the domain of integration as
$$
\align
\summ_{l=1}^s &(-1)^{s-1}\mu(\,dx_1)\dots\mu(\,dx_{l-1})
(\delta_{\xi_{j}}(\,dx_l)-\mu(\,dx_l))\mu(\,dx_{l+1})\dots\mu(\,dx_s) \\
&\qquad +(-1)^{s-1}(s-1)\mu(dx_1)\dots\mu(dx_s).
\endalign
$$
Here we have exploited that all other terms of this product disappear
in the domain of integration. Let us also observe that the term
$(-1)^{s-1}(s-1)\mu(\,dx_1)\dots\mu(\,dx_l)$ appears $n$-times as we
sum up for $1\le j\le n$. Similar calculation can be made for all
partitions $P\in\Cal P$ and all sets contained in the partitions,
only the notation of the indices will be more complicated.
 
Let us fix a general partition $P=\{R_1,\dots,R_u\}\in\Cal P$, and let
us rewrite the inner sum in formula (A2) in a more appropriate form.
We can get the proof of Proposition~3 by means of summing up the
identities we get in such a way for all $P\in\Cal P$. To get the
desired formula fix some vector $(j_1,\dots,j_k)$ such
that $H(j_1,\dots,j_k)=P$, and let us rewrite the multiple integral
in the inner sum of (A2) corresponding to this index.
 
We can get by working out the above mentioned calculation in the
general case that for a vector
$(j_1,\dots,j_k)$ such that $H(j_1,\dots,j_k)=P$ the relation
$$
\aligned
&\int' f(x_1,\dots,x_k) \(\delta_{\xi_{j_1}}(\,dx_1)-\mu(\,dx_1)\)
\dots \(\delta_{\xi_{j_k}}(\,dx_k)-\mu(\,dx_k)\)  \\
&\qquad=\sum_{V\in\Cal T(P)}\alpha(V,P)\int f(x_1,\dots,x_k)
\prod_{s\in V}
\(\delta_{\xi_{j_s}}(\,dx_s)-\mu(\,dx_s)\) \prod_{s'\in\{1,\dots,k\}
\setminus V} \mu(\,dx_{{s'}})
\endaligned \tag A3
$$
holds with some appropriate constants $\alpha(V,P)$, where the class
$\Cal T(P)$ which consists of subsets of $\{1,\dots,k\}$ and depends
on the partition $P=\{R_1,\dots,R_u\}$ is defined in the following
way. For all elements $R_t$, $1\le t\le u$, of the partition~$P$ a
set $V\in\Cal T(P)$ contains zero or 1 elements of the set $R_t$. If
$R_t=\{b_t\}$ consists of one elements, then the set $V$ contains
this point $b_t$. $\Cal T(P)$ consists of all subsets of
$\{1,\dots,k\}$ which satisfy the two above properties.
 
The coefficients $\alpha(V,P)$ at the right-hand side of~(A3) could
be calculated explicitly, but we do not have to do this. It is
enough to know that it depends only on the partition $P$ and the set
$V\in\Cal T(P)$. Let us also observe that at the right-hand side
of~(A3) the prime is missing in the integral, i.e. here integration
is taken on the whole space $X^k$, the diagonals are not taken out
from the domain of integration. Indeed, it can be seen that because
of the non-atomic property  of the measure $\mu$ we do not change
the value of the integrals at the right-hand side of~(A3) by inserting
the diagonals to the domain of integration.
 
Formula (A3) can be rewritten in the following way.
$$ \allowdisplaybreaks
\align
&\int' f(x_1,\dots,x_k) \(\delta_{\xi_{j_1}}(\,dx_1)-\mu(\,dx_1)\)
\dots \(\delta_{\xi_{j_k}}(\,dx_k)-\mu(\,dx_k)\)  \\
&\qquad=\sum_{V\in\Cal T(P)} \alpha(V,P)
\(\(\prod_{s'\in\{1,\dots,k\}\setminus V} P_{\mu,s'}
\prod_{s\in V} Q_{\mu,s}\)\) f(\xi_{j_s},s\in V).  \tag A4
\endalign
$$
Here $Q_{\mu,s}=I-P_{\mu,s}$ is the operator $Q_\mu$ defined in~(2.8)
if $Y_1$ is the product of the first $s-1$ components of the product
space $X^k$, $Z$ is its $s$-th component and $Y_2$ is
the product of the last $k-s$ components. The operator $P_{\mu,s'}$ is
the operator $P_\mu$ defined in (2.7) with the choice of $Y_1$ as
the first $s'-1$ components, $Z$ as the $s'$-th component and $Y_2$
as the product of the last $k-s'$ components of the space $X^k$. To
see why formula (A4) holds we have to understand that integration
with respect to $\(\delta_{\xi_{j_s}}(\,dx_s)-\mu(\,dx_s)\)$ means
the application of the operator $Q_{\mu,s}$ and then putting the
value $\xi_{j_s}$ in the argument $x_s$, while integration with
respect to $\mu(\,dx_{s'})$ means the application of the operator
$P_{\mu,s'}$. Beside this, the operators $Q_{\mu,s}$ and
$P_{\mu,s'}$ are exchangeable.
 
By fixing some $V\in\Cal T(P)$ and summing up the term corresponding to
it at the right-hand side of formula~(A4) for all $(j_1,\dots,j_k)$
such that $H(j_1,\dots,j_k)=P$ we get that
$$
\alpha(V,P) \!\!\!\!\!\!  \sum\Sb (j_1,\dots,j_k)\\ 1\le j_s\le n,
\;1\le s\le k\\ H(j_1,\dots,j_k)=P\endSb
\(\prod_{s'\in\{1,\dots,k\}\setminus V}  \!\!\!\! P_{\mu,s'}
\prod_{s\in V} Q_{\mu,s}\) f(\xi_{j_s},\,s\in V)
=\bar\alpha(V,P,k,n)I_{n,|V|}(f_V) \tag  A5
$$
where $I_{n,|V|}(f_V)$ is the $U$-statistic of order $|V|$ with the
kernel function
$$
f_V(x_s, s\in V)=\(\prodd_{s'\in\{1,\dots,k\}\setminus V}
P_{\mu,s'} \prodd_{s\in V} Q_{\mu,s}\)f(x_1,\dots,x_k), \tag A6
$$
and the coefficients $\bar\alpha(V,P,k,n)$ at the right-hand side of
(A5) are appropriate coefficients which could be calculated
explicitly. But we do not need such a formula. It can be shown with
some work that they satisfy the inequality $|\bar\alpha(V,P,k,n)|\le
D(k) n^{\beta(P,V)}$, where $\beta(P,V)=u-|V|$
is the number of those components $R_s$, $1\le s\le u$, of the
partition $P$ for which $R_s\cap V=\emptyset$, and the constant
$D(k)<\infty$ depends only on the multiplicity~$k$ of the integral
$J_{n,k}(f)$. Such an estimate is sufficient for us.
 
We get from relations (A2), (A4) and (A5) by summing up identity (A5)
for all $P\in\Cal P$ and $V\in\Cal T(P)$ that
$$
J_{n,k}(f)=\sum_{V\subset \{1,2,\dots,k\}} C(n,k,V)  n^{-|V|/2}
I_{n,|V|}(f_V) \tag A7
$$
with some coefficients $C(n,k,V)$. Moreover, a careful analysis
shows that the above coefficients satisfy the inequality
$|C(n,k,V)|\le G(k)$ with some constant $G(k)>0$. The explicit
expression for the coefficients $C(n,k,V)$ has a rather complicated
form, but the above estimate about their magnitude is sufficient
for our purposes. This estimate for $C(k,n,V)$ is sharp, because for a
fixed set $V$ those partitions $P\in\Cal P$ which contain the $|V|$
one-point subsets of $V$ and $(k-|V|)/2$ subsets of cardinality 2 of
$\{1,\dots,k\}\setminus V$ yield a contribution of order
$n^{-k/2}n^{k/2-|V|/2}$ to the coefficient $C(n,k,V)n^{-|V|/2}$.
A more careful analysis also shows that for a fixed set
$V\subset\{1,\dots,k\}$ the sequence $C(n,k,V)$ has a finite limit
as $n\to\infty$. It is not difficult to see that
$C(n,k,\{1,\dots,k\})=1$ for $V=\{1,\dots,k\}$.
 
The definition of the function $f_V$ in formula (A6) agrees with
the definition of $f_V$ in formula (2.11). Hence formulas~(A6),~(A7)
and the estimates on the coefficients $C(n,k,V)$ in formula~(A7)
imply Proposition~3.
 
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Springer--Verlag, New York
 
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Supported by the OTKA foundation Nr. 037886
 
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