
\beginsection 11. The diagram formula for products of degenerate
$U$-statistics.

There is a natural analog of the diagram formula for the products
of  Wiener--It\^o integrals both for the products of multiple 
integrals with respect to normalized empirical measures and for 
the products of degenerate $U$-statistics. These two results are 
closely related. They express the products of multiple random 
integrals or of degenerate $U$-statistics as a sum of multiple 
random integrals or degenerate $U$-statistics respectively. In 
this work the diagram formula for multiple integrals with 
respect to a normalized empirical measure will be discussed 
only at an informal level, while a complete proof of the 
analogous result about degenerate $U$-statistics will be given. 
The reason for such an approach is that the diagram formula for 
the product of degenerate $U$-statistics is more useful in the 
study of the problems discussed in this work.

We want to get good estimates about the high moments both of
multiple random integrals and of degenerate $U$-statistics. In the 
case of degenerate $U$-statistics the diagram formula yields an 
explicit formula for these moments. It expresses the product whose 
expected value has to be calculated as a sum of degenerate
$U$-statistics of different order. Beside this the expected
value of all degenerated $U$-statistics of order $k\ge1$ equals
zero. Hence the expected value we are interested in equals the
sum of the zero order terms appearing in the diagram formula.

The analogous problem about the moments of multiple integrals 
with respect to a normalized empirical measure is more difficult.
The diagram formula enables us to express these moments as the
sum of the expectation of multiple random integrals of different
order also in this case.  But the expected value of random 
integrals of order $k\ge1$ with respect to a normalized empirical 
distribution may be non-zero. It was shown in an example presented 
before the proof of Theorem~9.4 that this is really possible. 

First I give an informal description of the diagram formula for
the product of two random integrals with respect to a normalized
empirical measure. Its analog, the diagram formula for the product
of two Wiener--It\^o integrals can be described in an informal way 
by means of formulas (10.13) and (10.13a) together with the `identity'
$\mu_W(\,dx)^2=\mu(\,dx)$ in their interpretation. The diagram
formula for the product of two multiple integrals with respect to
a normalized empirical measure has a similar representation. 
(Observe that in the definition of the random integral 
$J_{n,k}(\cdot)$ given in formula~(4.8) the diagonals are omitted
from the domain of integration, similarly to the case of 
Wiener--It\^o integrals.)  In this case such a version of 
formulas~(10.13) and~(10.13a) can be applied, where the random 
integrals $Z_{\mu,k}$ are replaced by $J_{n,k}$, and the white 
noise measures $\mu_W$ are replaced by the normalized empirical 
measures $\nu_n=\sqrt n(\mu_n-\mu)$. But the analog of the 
`identity' $\mu_W(\,dx)^2=\mu(\,dx)$ needed in the interpretation 
of these formulas has a different form. Namely, it states that 
$\nu_n(\,dx)^2=\mu(\,dx)+\frac1{\sqrt n}\nu_n(\,dx)$. 
Let us `prove' this new `identity'.

Take a small set $\Delta$, i.e. a set $\Delta$ such that
$\mu(\Delta)$ is very small, write down the identity
$\nu_n(\Delta)^2=n\mu_n(\Delta)^2+n\mu(\Delta)^2
-2n\mu_n(\Delta)\mu(\Delta)$, and observe that only a second order
error is committed if the terms $n\mu(\Delta)^2$ and
$2n\mu_n(\Delta)\mu(\Delta)$ are omitted at the right-hand side
of this identity. Moreover, also a second order error is committed
if $n\mu_n(\Delta)^2$ is replaced by $\mu_n(\Delta)$, because it
has second order small probability that there are at least two
sample points in the small set $\Delta$. On the other
hand, $n\mu_n(\Delta)^2=\mu_n(\Delta)$ if $\Delta$ contains only
zero or one sample point. The above considerations suggest that
$\nu_n(\,dx)^2=\mu_n(\,dx)=\mu(\,dx)+\frac1{\sqrt n}
[\sqrt n(\mu_n(\,dx)-\mu(\,dx))]=\mu(\,dx)+\frac1{\sqrt n}\nu_n(\,dx)$.
(This means that in the `identity' expressing the square 
$\nu_n(\,dx)^2$ of a normalized empirical measure a correcting term 
$\frac1{\sqrt n}\nu_n(\,dx)$ appears. If the sample size~$n\to\infty$,
then the normalized empirical measure tends to a white noise with
counting measure~$\mu$, and this correcting term disappears.)

In paper [32] the diagram formula for the product of two multiple
integrals with respect to a normalized empirical measure is proved
with a different notation. It says that the identity suggested by
the above heuristic argument really holds. This result may also help 
in the proof of the diagram formula for degenerate $U$-statistics.
But a direct proof of this result seems to be simpler.

In the proof of the diagram formula for the product of two
degenerate $U$-statistics first we write this product as the sum 
of (not necessarily degenerate) $U$-statistics. Then by applying 
Hoeffding's decomposition for each term in this sum the product 
of two degenerate $U$-statistics can also be written as a sum of 
degenerate $U$-statistics. Actually we apply a slightly refined 
version of the Hoeffding decomposition where we exploit that we 
took the product of two {\it degenerated}\/ $U$-statistics. Such 
a calculation yields the diagram formula for the product of two 
degenerate $U$-statistics. With the help of a good notation and 
some additional work also the product of several degenerate 
$U$-statistics can be written as the sum of appropriate 
degenerate $U$-statistics. In such a way we get the general form 
of the diagram formula for the product of degenerate 
$U$-statistics. 

In this section I formulate the diagram formula for the product of 
two and finitely many degenerate $U$-statistics together with an
estimate about the $L_2$-norm of the kernel functions of the
degenerate $U$-statistics appearing in the diagram formula, and a
formula about the expectation of products of degenerate 
$U$-statistics. To formulate these results some new notations have 
to be introduced. The proofs of the results in this section are 
postponed to the next section.

In the formulation of the diagram formula for the product of
degenerate $U$-statistics a more general class of diagrams have 
to be considered than in the case of multiple Wiener--It\^o 
integrals. We shall define these new diagrams under the name 
coloured diagrams. Tthe kernel functions of the $U$-statistics 
appearing in the diagram formula will be introduced with their 
help.

A class of coloured diagrams $\Gamma(k_1,\dots,k_m)$ will be 
defined whose vertices will be the pairs $(p,r)$, $1\le p\le m$, 
$1\le r\le k_p$, and the set of vertices $(p,r)$, $1\le r\le k_p$, 
with a fixed number $p$ will be called the $p$-th row of the diagram. 
To define the coloured diagrams of the class $\Gamma(k_1,\dots,k_m)$ 
first the notions of chain and coloured chain will be introduced. A 
sequence $\beta=\{(p_1,r_1),\dots,(p_s,r_s)\}$ with 
$1\le p_1<p_2<\dots<p_s\le m$ and $1\le r_u\le k_{p_u}$ for all 
$1\le u\le s$ will be called a chain. The number $s$ of the pairs 
$(p_u,r_u)$ in this sequence, denoted by $\ell(\beta)$, will be 
called the length of the chain $\beta$. Chains of length 
$\ell(\beta)=1$, i.e. chains consisting only of one element 
$(p_1,r_1)$ are also allowed. We shall define a function 
$c(\beta)=\pm1$ which will be called the colour of the 
chain~$\beta$, and the pair $(\beta,c(\beta))$ will be called a
coloured chain. We shall allow arbitrary colouring $c(\beta)=\pm1$
of a chain with the only restriction that a chain of length~1 
can only get the colour $-1$, i.e. $c(\beta)=-1$ if $\ell(\beta)=1$.

A coloured diagram $\gamma\in\Gamma(k_1,\dots,k_m)$, 
$\gamma=\{\beta(l_1),\dots,\beta(l_s)\}$ is a partition of the set 
$\{(p,r)\colon\; 1\le p\le m,\, 1\le r\le k_p\}$ to the union
of some coloured chains $\beta(l_1),\dots,\beta(l_s)$, i.e. each 
vertex $(p,r)$ is the element of exactly one chain 
$\beta(l_j)\in\gamma$. Beside this, each chain $\beta(l_j)$ of a 
diagram $\gamma$ has a colour $c_\gamma(\beta(l_j))=\pm1$. The set 
$\Gamma(k_1,\dots,k_m)$ consists of all partitions of the set of
vertices $\{(p,r), 1\le p\le m,\,1\le r\le k_p\}$ to coloured 
chains, where an arbitrary colouring of the chains with the 
numbers~$\pm1$ is allowed with the only restriction that for a 
chain $\beta\in\gamma$ of length $\ell(\beta)=1$ of a diagram 
$\gamma\in\Gamma(k_1,\dots,k_m)$ $c_\gamma(\beta)=-1$. In our 
notation  we have introduced an indexation 
$\beta(l_s)=\beta(l_s,\gamma)$, $1\le l_1<l_2<\cdots<l_s$, of 
the chains of a coloured diagram $\gamma\in\Gamma(k_1,\dots,k_m)$.
Both the number~$s$ and the indices~$l_1,\dots,l_s$ may depend 
on~$\gamma$. Such a notation will be useful in our later 
considerations. It also turned out useful to allow more 
general indexation of these chains with numbers~$l_1,\dots,l_s$ 
and not only with the numbers~$1,\dots,s$.

We shall also introduce an enumeration of the vertices of a 
coloured diagram $\gamma\in\Gamma(k_1,\dots,k_m)$ with the help
of the enumeration of its chains. Given a coloured diagram 
$\gamma=(\beta(l_1),\dots,\beta(l_s))\in\Gamma(k_1,\dots,k_m)$
we define the indices $\alpha_\gamma(p,r)$ of a vertex~$(p,r)$ 
of this diagram by the formula $\alpha_\gamma(p,r)=l_j$ if 
$(p,r)\in\beta(l_j)$. We shall divide the set of indices 
$\{l_1,\dots,l_s\}$ of the chains contained in a coloured 
diagram~$\gamma$ into two disjoint sets 
$O(\gamma)=\{l_j\colon\; 1\le j\le s,\, c_\gamma(\beta(l_j))=-1\}$,
called the set of open indices of the diagram~$\gamma$ and 
$C(\gamma)=\{l_j\colon\; 1\le j\le s,\, c_\gamma(\beta(l_j))=1\}$,
called the set of closed indices of the diagram~$\gamma$.
We shall also list the elements of $O(\gamma)$ in an increasing 
order, i.e. write 
$O(\gamma)=\{\bar l_1,\dots,\bar l_{|O(\gamma)|}\}$, 
$\bar l_1<\bar l_2<\cdots<\bar l_{|O(\gamma)|}$. (We shall denote
the cardinality of a finite set~$A$ by~$|A|$ in the sequel.)
We defined the coloured diagrams and introduced their open and 
closed indices, because, as we shall see, in the diagram formula 
such degenerate $U$-statistics appear whose kernel functions are 
defined with the help of these coloured diagrams, and the indices 
of the arguments of the kernel function corresponding to the 
coloured diagram~$\gamma$ are closely related to the chains 
of~$\gamma$ with colour~$-1$, hence to the open indices of~$\gamma$.

In the diagram formula we express the product 
$\prodd_{p=1}^m I_{n,k_p}(f_p)$ of degenerate $U$-statistics with
canonical kernel functions~$f_p$ of $k_p$ variables as the sum of
appropriate degenerate $U$-statistics. The kernel functions
of the degenerate $U$-statistics appearing in this representation of 
the product of degenerate $U$-statistics will depend on the above 
defined coloured diagrams $\gamma$, and they will be denoted by 
$F_\gamma$, $\gamma\in\Gamma(k_1,\dots,k_m)$. In the definition of 
these functions~$F_\gamma$ we shall apply the operators introduced 
below.

Given a function $h(x_{u_1},\dots,x_{u_r})$ with coordinates in the
space $(X,\Cal X)$ (the indices $u_1,\dots,u_r$ are all different,
otherwise they can be chosen in an arbitrary way) and a probability
measure~$\mu$ on the space $(X,\Cal X)$ let us introduce
its transforms $P_{u_j}h$ and $Q_{u_j}h$, $1\le j\le r$, by
the formulas
$$ 
(P_{u_j}h)(x_{u_l}\colon\, u_l\in \{u_1,\dots,u_r\}\setminus \{u_j\})
=\int h(x_{u_1},\dots,x_{u_r})\mu(\,dx_{u_j}), \quad 1\le j\le r,
\tag11.1 
$$
and
$$
(Q_{u_j}h)(x_{u_1},\dots,x_{u_r})=h(x_{u_1},\dots,x_{u_r})-
\int h(x_{u_1},\dots,x_{u_r})\mu(\,dx_{u_j}), \quad 1\le j\le r.
\tag11.2
$$
(These formulas are very similar to the definition of the operators
$P_j$ and $Q_j$ introduced in formula~(9.1) before the proof of the 
Hoeffding decomposition.)

First we consider the product of two degenerate $U$-statistics, i.e. 
the case $m=2$. Let us have a measurable space $(X,\Cal X)$ with a 
probability measure~$\mu$ on it together with two measurable 
functions $f_1(x_{1},\dots,x_{k_1})$ and $f_2(x_1,\dots,x_{k_2})$ 
of $k_1$ and $k_2$ variables on this space which are canonical with
respect to the measure $\mu$. Let $\xi_1,\xi_2,\dots$ be a sequence 
of $(X,\Cal X)$ valued, independent and identically distributed 
random variables with distribution~$\mu$. We want to express the 
product $I_{n,k_1}(f_1)I_{n,k_2}(f_2)$ of degenerate $U$-statistics 
defined with the help of the above random variables and kernel 
functions~$f_1$ and~$f_2$ as a sum of degenerate $U$-statistics.
For this goal we introduce some notations. 

Given two functions $f_1(x_1,\dots,x_{k_1})$ and 
$f_2(x_1,\dots,x_{k_2})$ and a coloured diagrams 
$\gamma\in\Gamma(k_1,k_2)$ consisting of $s$ coloured chains 
$\beta(l_1),\dots,\beta(l_s)$ we define the function
$$
\overline{(f_1\circ f_2)}_\gamma(x_{l_1},\dots,x_{l_{s}})=
f_1(x_{\alpha_\gamma(1,1)},\dots,x_{\alpha_\gamma(1,k_1)})
f_2(x_{\alpha_\gamma(2,1)},\dots,x_{\alpha_\gamma(2,k_2)}), 
\tag11.3
$$
where $\alpha_\gamma(p,r)$ denotes the index of the vertex $(p,r)$
of the diagram $\gamma$ in their above defined 
enumeration~$\alpha_\gamma$. (In formula~(11.3) all arguments of 
the functions $f_1$ and $f_2$ have different indices. But the indices 
$\alpha_\gamma(1,j)$ and $\alpha_\gamma(2,j')$ may agree for some
pairs $(j,j')$. This happens if the vertices $(1,j)$ and $(2,j')$ 
belong to the same  chain $\beta\in\gamma$ of length~2.) Let us also 
define the function
$$
(f_1\circ f_2)_\gamma(x_{l_p},\;l_p\in O(\gamma)) 
=\(\prod_{p\in C(\gamma)} P_{p}\prod_{p\in O_2(\gamma)} Q_p \)
\overline{(f_1\circ f_2)}_\gamma(x_{l_1},\dots,x_{l_s}),  \tag11.4
$$
with the operators $P_p$ and $Q_p$ defined (with a different indexation)
in formulas~(11.1) and~(11.2), where $C(\gamma)$ is the set of indices
of the closed diagrams of~$\gamma$, and $O_2(\gamma)\subset O(\gamma)$, 
defined as $O_2(\gamma)=\{l\colon\; 
c_\gamma(\beta_l)=-1, \text{ and }\ell(\beta(l))=2\}$, is the set of
indices of the chains of~$\gamma$ with colour~$-1$ and length~2.
are the above defined sets of open and closed indices of the 
diagram~$\gamma$. The arguments of the 
function $(f_1\circ f_2)_\gamma$ are the indices of the open 
vertices of the diagram~$\gamma$. Let us also remark that the 
operators $P_p$ and $Q_p$ in formula~(11.4) are exchangeable, hence 
it is not important in what order we apply them.

The function $F_\gamma(f_1,f_2)$ we apply in the formulation of 
the diagram formula in the special case when the product of two 
degenerate $U$-statistics is considered is similar to the function 
$(f_1\circ f_2)_\gamma$ introduced in~(11.4). We need a small 
technical step for its definition. We want to work with such a 
function whose variables are indexed with the numbers 
$1,2,\dots,|O(\gamma)|$ while the indices of the function 
$f_1\circ f_2)_\gamma$ are the elements of the set 
$O(\gamma)=\{\bar l_1,\dots,\bar l_{|O(\gamma)|}\}$. Hence we 
define the function $t=t_\gamma$ on the set $O(\gamma)$ by the 
formula $t(\bar l_j)=j$, $1\le j\le|O(\gamma)|$, and introduce 
the function
$$
F_\gamma(f_1,f_2)(x_1,x_2,\dots,x_{|O(\gamma)|})=
(f_1\circ f_2)_\gamma(x_{t(l_p)},\;l_p\in O(\gamma)). \tag11.5
$$
Let me remark that for different enumerations 
$\beta(l_1),\dots,\beta(l_s)$ of the chains of a coloured diagram 
$\gamma$ the function $F_\gamma(f_1,f_2)$ we defined by formulas 
(11.1)--(11.5) may be slightly different. One of them can be 
obtained by reindexing the variables $x_1,\dots,x_{|O(\gamma)|}$ 
in these functions. But the value of the $U$-statistic 
$I_{n,|O(n)|}(F_\gamma(f_1,f_2))$ does not depend on the 
indexation of the variables in its kernel function, hence on the 
enumeration of the chains of~$\gamma$. For a similar reason
the value of $I_{n,|O(n)|}(F_\gamma(f_1,f_2))$ depends only on
the cardinality of $|O(\gamma)|$, $|O_2(\gamma)|$ and $|C(\gamma)|$
of the coloured diagram $\gamma$, and also a reindexation of the 
arguments of $f_1$ or $f_2$ does not change the value of the 
$U$-statistic $I_{n,|O(n)|}(F_\gamma(f_1,f_2))$.

Next I formulate the diagram formula for the product of two 
degenerate $U$-statistics with the help of the above defined 
quantities. 

\medskip\noindent
{\bf Theorem 11.1. (The diagram formula for the product of two
degenerate $U$-statistics).} {\it Let a sequence of independent
and identically distributed random variables $\xi_1,\xi_2,\dots$
be given with some distribution $\mu$ on a measurable space
$(X,\Cal X)$  together with two bounded canonical functions
$f_1(x_1,\dots,x_{k_1})$ and $f_2(x_1,\dots,x_{k_2})$ with respect 
to the probability measure~$\mu$ on  the product spaces 
$(X^{k_1},\Cal X^{k_1})$ and $(X^{k_2},\Cal X^{k_2})$ respectively. 
Let us take the class of coloured diagrams $\Gamma(k_1,k_2)$ 
introduced above together with the functions $F_\gamma(f_1,f_2)$ 
defined in formulas (11.1)---(11.5).

For all $\gamma\in\Gamma$ $F_\gamma(f_1,f_2)_\gamma$ is a canonical 
function with respect to the measure $\mu$ with $|O(\gamma)|$ 
arguments, where $O(\gamma)$ and $C(\gamma)$ denote the set of open 
and closed indices of the diagram~$\gamma$. The product of the 
degenerate $U$-statistics $I_{n,k_1}(f_1)$ and 
$I_{n,k_2}(f_2)$, $n\ge\max(k_1,k_2)$, defined in (8.7) 
can be expressed as
$$
\aligned
&(n^{-k_1/2}k_1!I_{n,k_1}(f_1))(n^{-k_2/2}k_2! I_{n,k_2}(f_2)) \\
&\qquad =\sumn_{\gamma\in\Gamma(k_1,k_2)}\prodd_{j=1}^{|C(\gamma)|}
\(\frac{n-s(\gamma)+j}n\) n^{-W(\gamma)/2}\cdot
n^{-|O(\gamma)|/2}|O(\gamma)|!I_{n,|O(\gamma)|}(F_\gamma(f_1,f_2))
\endaligned \tag11.6
$$
with $W(\gamma)=k_1+k_2-|O(\gamma)|-2|C(\gamma)|$ and
$s(\gamma)=|O(\gamma)|+|C(\gamma)|$ (which equals the number of
coloured diagrams in~$\gamma$), where $\sum^{\prime(n)}$ means 
that summation is taken only for such coloured diagrams 
$\gamma\in\Gamma(k_1,k_2)$ which satisfy the inequality 
$s(\gamma)\le n$, and $\prodd_{j=1}^{|C(\gamma)|}$ 
equals 1 in the case $|C(\gamma)|=0$. The term 
$I_{n,|O(\gamma)|}(F_\gamma(f_1,f_2))$ can be replaced by
$I_{n,|O(\gamma)|}(\text{\rm Sym}F_\gamma(f_1,f_2))$ in
formula~(11.6).

Consider the $L_2$-norm of the functions $F_\gamma(f_1,f_2)$
defined by the formula
$$
\|F_\gamma(f_1,f_2)\|_2^2=\|(f_1\circ f_2)_\gamma\|_2^2=
\int (f_1\circ f_2)_\gamma^2(x_{l_p},\,l_p\in O(\gamma))
\prod_{l_p\in O(\gamma)}\mu(\,dx_{l_p}).
$$
The inequality
$$
\|F_\gamma(f_1,f_2)\|_2=\|(f_1\circ f_2)_\gamma\|_2\le \|f_1\|_2\|f_2\|_2
\quad \text{if }\; W(\gamma)=0 \tag11.7
$$
holds for this norm. The condition $W(\gamma)=0$ in formula~(11.7) 
means that the diagram $\gamma\in\Gamma(k_1,k_2)$ has no 
chains~$\beta$ of length $\ell(\beta)=2$ with 
colour~$c_\gamma(\beta)=-1$. In the case of a general diagram 
$\gamma\in\Gamma(k_1,k_2)$ the inequality
$$
\|F_\gamma(f_1,f_2)\|_2=\|(f_1\circ f_2)_\gamma\|_2
\le 2^{W(\gamma)}\min(\|f_1\|_2,\|f_2\|_2)
\tag11.8
$$
holds if the $L_\infty$-norm of  the functions $f_1$ and $f_2$ satisfies 
the inequalities $\|f_1\|_\infty\le1$ and $\|f_2\|_\infty\le1$. 
Relations~(11.7) and~(11.8) also hold for non-canonical 
functions~$f_1$ and~$f_2$.}

\medskip
Inequality (11.7) is actually a repetition of estimate (10.11)
about the diagrams appearing in the case of Wiener--It\^o integrals.
Inequality (11.8) yields a weaker bound about the $L_2$-norm
$\|F_\gamma(f_1,f_2)\|_2=\|(f_1\circ f_2)_\gamma\|_2$ for a 
general diagram $\gamma$. In particular, it depends not only on 
the $L_2$-norm, but also on the $L_\infty$-norm of the 
functions~$f_1$ and~$f_2$. This is closely related to the fact that 
in the estimates on the distribution of $U$-statistics, --- unlike 
the case of Wiener--It\^o integrals, --- a condition is imposed not 
only on the $L_2$-norm of the kernel  function~$f$, but also on its 
$L_\infty$-norm. I return to this question later.

\medskip\noindent
{\it Remark 1.}\/ The expression 
$W(\gamma)=k_1+k_2-|O(\gamma)|-2|C(\gamma)|$ appearing in 
formulas~(11.6), (11.7) and~(11.8) has the following content. It 
equals the number of those diagrams $\beta(l_j)\in\gamma$ for which
$\ell(\beta(l_j))=2$, and $c_\gamma(\beta(l_j))=-1$. Indeed, if 
$W(\gamma)$ denotes the number of such chains, and $\bar W(\gamma)$ 
equals the number of chains $\beta(l_j)\in\gamma$ for which 
$\ell(\beta(l_j))=1$ (and as a consequence $c_\gamma(\beta(l_j))=-1$),
then $W(\gamma)+\bar W(\gamma)=|O(\gamma)|$, and 
$2W(\gamma)+\bar W(\gamma)+2|C(\gamma)|=k_1+k_2$. These
identities imply the statement of this remark.

\medskip\noindent
{\it Remark 2.}\/ The term $I_{n,|O(\gamma)|}(F_\gamma(f_1,f_2))$
appeared in the sum at the right-hand side of~(11.6) only if the 
condition $s(\gamma)\le n$ was satisfied. This restriction in the
summation had a technical character, which has no great importance 
in our investigations. It is related to the fact that a 
$U$-statistic $I_{n,k}(f)$ exists only if $n\ge k$. As a 
consequence, some $U$-statistics disappear at the right-hand side 
of~(11.6) if the sample size~$n$ of the $U$-statistics is 
relatively small. The term $I_{n,|O(\gamma)|}(F_\gamma(f_1,f_2))$ 
appeared in~(11.6) through the Hoeffding decomposition of a 
$U$-statistic with kernel function 
$\overline{(f_1\circ f_2)}_\gamma$ defined in~(11.3). This 
function has $s(\gamma)$ arguments, and the $U$-statistic 
corresponding to it appears in our calculations only if the sample 
size~$n$ is not smaller than this number.

\medskip
Let us recall the convention introduced after the definition of 
canonical degenerate $U$-statistics by which $I_{n,0}(c)$ is a 
degenerate $U$-statistic of order zero, and $I_{n,0}(c)=c$ for a 
constant $c$. By applying this convention we write 
$F_\gamma((f_1, f_2)=f_1\circ f_2$ in relation~(11.6) for those 
diagrams~$\gamma$ for which $|O(\gamma)|=0$, i.e. 
$c_\gamma(\beta)=1$ for all chains $\beta\in\gamma$. We shall 
introduce another convention which implies that Theorem~11.1 is 
valid also in the degenerate case when the function~$f_{k_1}=c$ 
with a constant~$c$, and $k_1=0$. In this case $\Gamma(k_1,k_2)$ 
consists of only one diagram $\gamma$ containing the chains 
$\beta_j=\{j\}$ of length one and colour $c_\gamma(\{j\})=-1$, 
$1\le j\le k_2$. We define $I(F_\gamma(f_1,f_2))=cf_2$ in this 
case. Beside this, we have 
$W(\gamma)=k_1+k_2-|O(\gamma)|-2|C(\gamma)|=0$, $|O(\gamma)|=k_2$,
and $|C(\gamma)|=0$. Hence formula~(11.6) remains valid also in 
the case $k_1=0$. We have introduced this convention because the 
following inductive argument leading to the proof of the diagram 
formula for the product of degenerate $U$-statistics in the 
general case is valid under such a convention.

\medskip
Let us turn to the formulation of the general form of the diagram
formula for the product of degenerate $U$-statistics. First I
define a function $F_\gamma=F_\gamma(f_1,\dots,f_m)$ for each
coloured diagram $\gamma\in\Gamma(k_1,\dots,k_m)$ and collection 
of canonical functions (with respect to a probability measure~$\mu$ 
on a measurable space $(X,\Cal X)$) $f_1,\dots,f_m$ with $k_1$,\dots, 
and $k_m$ variables. These functions $F_\gamma$ will be the kernel 
functions of the degenerate~$U$-statistics at the right-hand side 
of the diagram formula. 

These functions $F_\gamma$ will be defined by induction with 
respect to the number~$m$ of the components in the product. For 
$m=2$ we have already defined the function $F_\gamma(f_1,f_2)$. 
Let the functions $F_\gamma(f_1,\dots,f_{m-1})$ be defined for 
each coloured diagram $\gamma\in\Gamma(k_1,\dots,k_{m-1})$. 
To define $F_\gamma(f_1,\dots,f_m)$ for a coloured diagram 
$\gamma\in\Gamma(k_1,\dots,k_m)$ first we define the predecessor 
$\gamma_{pr}=\gamma_{pr}(\gamma)\in\Gamma(k_1,\dots,k_{m-1})$ 
of~$\gamma$. We shall define the coloured diagram~$\gamma_{pr}$
together with an appropriate indexation of its element with the
help of the enumeration of the elements of~$\gamma$. Roughly 
speaking, the elements of~$\gamma_{pr}$ are the restrictions of 
the chains contained in~$\gamma$ to the first $m-1$ rows of the 
diagram, i.e. to the set 
$\{(p,r)\colon\;1\le p\le m-1,\,1\le r\le k_p\}$. But we must 
define also the colour of these resticted chains.

To define precisely the predecessor $\gamma_{pr}$ of $\gamma$ let 
us divide first the chains of the coloured diagram 
$\gamma=\{\beta(l_1),\dots,\beta(l_s)\}\in\Gamma(k_1,\dots,k_m)$ 
into two disjoint subsets $\gamma=\gamma_1\cup\gamma_2$, defined 
as $\gamma_1=\{\beta(l_j)\colon\;\beta(l_j)\in\gamma,
\,\beta(l_j)\cap\{(m,1),\dots,(m,k_m)\}\neq\emptyset\}$ and
$\gamma_2=\{\beta(l_j)\colon\;\beta(l_j)\in\gamma,
\,\beta(l_j)\cap\{(m,1),\dots,(m,k_m)\}=\emptyset\}$, i.e. a 
coloured chain $\beta\in\gamma$ belongs to $\gamma_1$ if it contains 
a vertex from the last row $\{(m,1),\dots,(m,k_m)\}$ of the diagram, 
and it belongs to $\gamma_2$ if it does not contain such a vertex. 
We define with the help of the chains $\beta(l_j)\in\gamma_1$ the
chains $\beta_{pr}(l_j)=\beta(l_j)\setminus\{(m,1),\dots,(m,k_m)\}$
and with the help of the chains $\beta(l_j)\in\gamma_2$
the chains $\beta_{pr}(l_j)=\beta(l_j)$. (For those chains 
$\beta(l_j)\in\gamma_1$ which consist only of one vertex of the 
form $(m,r)$, $1\le r\le k_m$, the corresponding chain 
$\beta_{pr}(l_j)$ would be the empty set. These empty sets 
are omitted from the set of chains $\beta_{pr}(l_j)\in\gamma_{pr}$.) 
The set of all above  defined chains $\beta_{pr}(l_j)$ provides 
a partition of the set of vertices 
$\{(p,r)\colon\; 1\le p\le m-1,\, 1\le r\le k_p\}$. The diagram
$\gamma_{pr}$ will consist of these chains $\beta_{pr}(l_j)$.
To complete the definition of the coloured diagram $\gamma_{pr}$ 
we still have to define the colour $c_{\gamma_{pr}}(\beta_{pr}(l_j))$ 
of these chains.

We define the colour of these chains by the formulas  
$c_{\gamma_{pr}}(\beta_{pr}(l_j))=-1$ if $\beta(l_j)\in\gamma_1$, 
and $c_{\gamma_{pr}}(\beta_{pr}(l_j))=c_\gamma(\beta(l_j))$ 
if $\beta(l_j)\in\gamma_2$. In such a way we defined the 
predecessor $\gamma_{pr}\in\Gamma(k_1,\dots,k_{m-1})$ of the
diagram $\gamma\in\Gamma(k_1,\dots,k_m)$. Moreover we gave an
indexation of the chains of $\gamma_{pr}$ with the help of the
indexation of the chains of~$\gamma$.  

With the help of the coloured diagram 
$\gamma_{pr}\in\Gamma(k_1,\dots,k_{m-1})$ we can define the function
$F_{\gamma_{pr}}=F_{\gamma_{pr}}(f_1,\dots,f_{m-1})$ which is a 
function of $|O(\gamma_{pr})|$ variables
$x_1,\dots,x_{|O(\gamma_{pr})|}$. We shall define the function 
$F_\gamma=F_\gamma(f_1,\dots,f_m)$  similarly to the definition of 
$F_\gamma(f_1,f_2)$ given by formulas~(11.3), (11.4) and~(11.5) in
the case $m=2$. In this case $F_{\gamma_{pr}}$ plays the role 
of the function~$f_1$ and $f_m$ the role of the function~$f_2$. To 
define the function $F_\gamma(f_1,\dots,f_m)$ we still have to 
define a coloured diagram 
$\gamma_{cl}=\gamma_{cl}(\gamma)\in\Gamma(|O(\gamma_{pr})|,k_m)$
that we shall call the closing diagram of $\gamma$. The heuristic
content of the diagram~$\gamma_{cl}$ is that it contains the
additional information we need to reconstruct the diagram
$\gamma\in\Gamma(k_1,\dots,k_m)$ if we know its 
predecessor~$\gamma_{pr}$. We shall define it together with an 
enumeration of its chains that depends on the enumeration 
of the chains of the diagram~$\gamma$.

To define the diagram $\gamma_{cl}$ let us first consider the listing
$O(\gamma_{pr})=\{\bar l_1,\dots,\bar l_{|O(\gamma_{pr})|}\}$,
$1\le \bar l_1<\bar l_2<\cdots<l_{|O(\gamma_{pr}|}$,
of the indices of the open indices of the diagram $\gamma_{pr}$
in increasing order. Let us fix a vertex $(1,j)$, 
$1\le j\le |O(\gamma_{pr})|$ in the first row of~$\gamma_{cl}$. 
We shall denote the chain of $\gamma_{cl}$ containing this 
vertex by $\beta_{cl}(\bar l_j)$, i.e. this chain get the 
index~$\bar l_j$,  and define it together with its colour in the 
following way. Let us consider the (open) chain 
$\beta_{pr}(\bar l_j)$ together with its `continuation' 
$\beta(\bar l_j)$. Clearly,
$\beta_{pr}(\bar l_j)\subset\beta(\bar l_j)$. If 
$\beta(\bar l_j)\in\gamma_1$, then
$\beta(\bar l_j)=\beta_{pr}(\bar l_j)\cup\{(m,r_j)\}$ with some 
integer $1\le r_j\le k_m$. In this case we define the chain 
containing the vertex $(1,j)$ as the diagram 
$\beta_{cl}(\bar l_j)=\{(1,j),(2,r_j)\}$ with this number~$r_j$,
and it gets the colour 
$c_{\gamma_{cl}}(\beta_{cl}(\bar l_j))=c_\gamma(\beta(\bar l_j))$.
If $\beta(\bar l_j)\in\gamma_2$, then 
$\beta_{pr}(\bar l_j)=\beta(\bar l_j)$, and we define the chain 
containing the vertex $(1,j)$ as the chain 
$\beta_{cl}(\bar l_j)=\{(1,j)\}$ of length~1 and with colour
$c_{\gamma_{cl}}(\beta_{cl}(\bar l_j))=-1$.

We still have to consider those vertices $(2,r)$ of
$\Gamma(|O(\gamma_{pr})|,k_m)$, $1\le r\le k_m$, for which there
exists a chain $\beta(l_{j(r)})\in\gamma$ such that 
$\beta(l_{j(r)})=\{m,r)\}$, because these are the vertices
of the set of vertices $\{(1,j)\colon\;1\le j\le|O(\gamma_{pr})|
\cup\{(2,r)\colon\;1\le r\le k_m\}$ which are not contained in the
previously defined chains $\beta_{cl}(\bar l_j)$. To cover these
vertices with an (appropriately indexed) chain of $\gamma_{cl}$ 
let us define the chains $\beta_{cl}(l_{j(r)})=\{(2,r)\}$ with 
the colour $c_{\gamma_{cl}}(\beta_{cl}(l_{j(r)}))=-1$ for 
such vertices~$(2,r)$. The above defined coloured chains provide a 
partition of the set $\{(1,j)\colon\;1\le j\le|O(\gamma_{pr})|
\cup\{(2,r)\colon\;1\le r\le k_m\}$,
and they are the elements of the coloured diagram~$\gamma_{cl}$.

We shall define the function $F_\gamma(f_1,\dots,f_m)$ with the help
of the above introduced diagrams $\gamma_{pr}$ and $\gamma_{cl}$
in the following way. Put, similarly to formula~(11.3),
$$
\aligned
&\overline{(F_{\gamma_{pr}}(f_1,\dots,f_{m-1})\circ f_m)}_\gamma
(x_{l_1},\dots,x_{l_s}) \\
&\qquad =F_{\gamma_{cl}}(x_{\alpha_{\gamma_{cl}}(1,1)},\dots,
x_{\alpha_{\gamma_{cl}}(1,|O(\gamma_{pr})|)})
f_m(x_{\alpha_{\gamma_{cl}}(2,1)},\dots,x_{\alpha_{\gamma_{pr}}(2,k_m)}), 
\endaligned \tag11.9
$$
where $s=s(\gamma_{cl})$ is the number of the chains contained 
in~$\gamma_{cl}$. The indices $l_1,l_2\dots$, and $l_s$ of the
variables at the left-hand side of~(11.9) agree with the indices 
of the chains of the diagram~$\gamma_{cl}$, and
$\alpha_{\gamma_{cl}}(p,r)$ denotes the index of the vertex $(p,r)$
of the diagram $\gamma_{cl}$ which is induced by the enumeration of
the indices of the chains in $\gamma_{cl}$. Next we define with the 
help of formula~(11.9), similarly to the relation~(11.4), the function
$$
\aligned
&(F_{\gamma_{pr}}(f_1,\dots,f_{m-1})
\circ f_m)_\gamma(x_p,\;p\in O(\gamma_{cl})) \\
&\qquad =\(\prod_{p\in C(\gamma_{cl})} P_{p}\prod_{p\in O_2(\gamma_{cl})} 
Q_p\)\overline{(F_{\gamma_{pr}}(f_1,\dots,f_{m-1}\circ f_m)}_\gamma
(x_p,\;p\in O(\gamma_{cl})\cup C(\gamma_{cl})) 
\endaligned \tag11.10
$$
with the operators $P_p$ and $Q_p$ defined (with a different indexation)
in formulas~(11.1) and~(11.2), where the sets $O(\gamma_{cl})$ and 
$C(\gamma_{cl})$ are the sets of open and closed 
indices of the diagram~$\gamma_{cl}$, and the set $O_2(\gamma_{cl})$
(for a general diagram with two rows) was defined after 
formula~(11.4). The function 
$(F_{\gamma_{pr}}(f_1,\dots,f_{m-1})\circ f_m)_\gamma$ depends only on the 
arguments indexed by the open indices of the diagram~$\gamma_{cl}$. 

The function $F_\gamma(f_1,\dots,f_m)$ will be defined by means of
a reindexation of the arguments of the function 
$(F_{\gamma_{pr}}(f_1,\dots,f_{m-1})\circ f_m)_\gamma
(x_{l_p},\;l_p\in O(\gamma_{cl}))$
which will be made to get a function with arguments 
$x_1,x_2,\dots,x_{|O(\gamma_{cl})|}$. It is defined, similarly to 
formula~(11.5), as
$$
F_\gamma(f_1,\dots,f_m)(x_1,x_2,\dots,x_{|O(\gamma_{cl})|})=
(F_{\gamma_{pr}}(f_1,\dots,f_{m-1})
\circ f_m)_\gamma(x_{t(l_p)},\;l_p\in O(\gamma_{cl})),
\tag11.11
$$
where the indices $t(l_p)$ are defined in the following way. We list
the open indices of the diagram $\gamma_{cl}$ in an increasing order 
as $O(\gamma_{cl})=\{\bar l_1,\dots,\bar l_{|O(\gamma_{cl})|}\}$, 
$\bar l_1<\bar l_2<\cdots<\bar l_{|O(\gamma_{cl})|}$, and define
the function $t(\cdot)$ on the set $O(\gamma_{cl})$ as $t(\bar l_p)=p$
for $1\le p\le |O(\gamma_{cl})|$. 

To complete the definition of the function $F_\gamma(f_1,\dots,f_m)$
observe that $|O(\gamma_{cl})|=|O(\gamma)|$. (Even the sets 
$O(\gamma_{cl})$ and $O(\gamma)$ agree with the enumeration of the
chains of these two diagrams we have chosen.) Hence we can write
$$
F_\gamma(f_1,\dots,f_m)(x_1,x_2,\dots,x_{|O(\gamma_{cl})|})=
F_\gamma(f_1,\dots,f_m)(x_1,x_2,\dots,x_{|O(\gamma)|}). \tag11.12
$$
Let me remark, that, just as in the case $m=2$, also in the case 
$m\ge2$ the value of the $U$-statistic 
$I_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m))$ does not depend on the 
enumeration of the chains of the coloured diagram~$\gamma$.


To formulate the general form of the diagram formula for the product
of degenerate $U$-statistics we introduce some quantities which will 
be the version  of the quantities appearing in the coefficients of 
the right-hand side of~(11.6) in Theorem~11.1. Put
$$
W(\gamma)=\sum_{l_p\in O(\gamma)}(\ell(\beta(l_p))-1)
+\sum_{l_p\in C(\gamma)}(\ell(\beta(l_p))-2),\quad
\gamma\in\Gamma(k_1,\dots,k_m), \tag11.13
$$
where $\ell(\beta)$ denotes the length of the chain~$\beta$.

To define the next quantity we need let us first introduce the
following notation. Given a chain 
$\beta=\{(p_1,r_1),\dots,(p_l,r_l)\}$,
$1\le p_1<p_2<\cdots<p_l\le m$, in the set 
$\{(p,r)\colon\;1\le p\le m,\,1\le r\le k_p\}$ let us define its 
upper level $u(\beta)=p_1$, and its deepest level $d(\beta)=l_p$. 
Let us define with their help for all diagrams 
$\gamma\in\Gamma(k_1,\dots,k_m)$ and integers~$p$, $1\le p\le m$,
the sets $\Cal B_1(\gamma,p)=\{\beta\colon\;\beta\in\gamma,\,
c_\gamma(\beta)=1,\,d(\beta)=p\}$, and $\Cal B_2(\gamma,p)
=\{\beta\colon\;\beta\in\gamma,\,c_\gamma(\beta)=-1,\,d(\beta)\le p\}
\cup\{\beta\colon\;\beta\in\gamma,\,u(\beta)\le p,\,d(\beta)>p\}$, 
i.e. $\Cal B_1(\gamma,p)$ consists of those chains $\beta\in\Gamma$ 
which have colour~$1$, all their vertices are in the first~$p$ 
rows of the diagram, and contain a vertex in the~$p$-th row, while 
$\Cal B_2(\gamma,p)$ consists of those chains $\beta\in\gamma$ 
which have either colour~$-1$, and all their vertices are in the 
first~$p$ rows of the diagram, or they have (with an arbitrary 
colour) a vertex both in the first~$p$ rows both in the remaining 
rows of the diagram. Put $B_1(\gamma,p)=|\Cal B_1(\gamma,p)|$ and
$B_2(\gamma,p)=|\Cal B_2(\gamma,p)|$. 
With the help of these numbers we define
$$
J_n(\gamma,p)=\left\{ 
\aligned
\prodd_{j=1}^{B_1(\gamma,p)}
&\(\frac{n-B_1(\gamma,p)-B_2(\gamma,p)+j}n\) 
\quad\text{if } B_1(\gamma,p)\ge1\\ 
& \!\!\!\!\!\!\!\!\!   1\quad \text{if } B_1(\gamma,p)=0
\endaligned \right. \tag11.14
$$
for all $2\le p\le m$ and diagrams $\gamma\in\Gamma(k_1,\dots,k_m)$. 

Theorem 11.2 will be formulated with the help of the above notations.

\medskip\noindent
{\bf Theorem 11.2. (The diagram formula for the product of several
degenerate $U$-statistics).} {\it Let a sequence of independent and
identically distributed random variables $\xi_1,\xi_2,\dots$ be
given with some distribution $\mu$ on a measurable space
$(X,\Cal X)$ together with $m\ge2$ bounded functions
$f_p(x_1,\dots,x_{k_p})$ on the spaces $(X^{k_p},\Cal X^{k_p})$,
$1\le p\le m$, canonical with respect to the probability
measure~$\mu$. Let us consider the class of coloured diagrams
$\Gamma(k_1,\dots,k_m)$ together with the functions
$F_\gamma=F_{\gamma}(f_1,\dots,f_m)$, 
$\gamma\in\Gamma(k_1,\dots,k_m)$, defined in formulas
(11.9)---(11.12) and the constants $W(\gamma)$ and  $J_n(\gamma,p)$, 
$1\le p\le m$, given in formulas~(11.13) and~(11.14).

The functions $F_\gamma(f_1,\dots,f_m)$ are canonical with respect 
to the measure $\mu$ with $|O(\gamma)|$ variables, and the product 
of the degenerate $U$-statistics $I_{n,k_p}(f_p)$, $1\le p\le m$, 
$n\ge \max\limits_{1\le p\le m} k_p$, defined in~(8.7) can be 
expressed as
$$
\aligned
\prod_{p=1}^m n^{-k_p/2}k_p!I_{n,k_p}(f_{k_p}) 
&=\sumnl_{\gamma\in\Gamma(k_1,\dots,k_m)} 
\(\prod_{p=2}^m J_n(\gamma,p)\) n^{-W(\gamma)/2} \\
&\qquad\qquad n^{-|O(\gamma)|/2} |O(\gamma)|!
I_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m)),
\endaligned\tag11.15
$$
where $\sum^{\prime(n,\,m)}$ means that summation is taken
for those $\gamma\in\Gamma(k_1,\dots,k_m)$ which satisfy the
relation $B_1(\gamma,p)+B_2(\gamma,p)\le n$ for all $2\le p\le m$ 
with the quantities $B_1(\gamma,p)$ and $B_2(\gamma,p)$ 
introduced before the definition of $J_n(\gamma,p)$ in~(11.14), and
the expression $W(\gamma)$ was defined in~(11.13).
The terms $I_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m))$ at the 
right-hand side of formula (11.15) can be replaced by 
$I_{n,|O(\gamma)|}(\text{\rm Sym}\,F_\gamma(f_1,\dots,f_m))$.}

\medskip
In Theorem 11.2 the product of such degenerate $U$-statistics were
considered, whose kernel functions were bounded. This also implies
that all functions $F_\gamma$ appearing at the right-hand side of
(11.15) are well-defined (i.e. the integrals appearing in their
definition are convergent) and bounded. In the applications of
Theorem~11.2 it is useful to have more information about the
behaviour of the functions $F_\gamma$. We shall need some good 
bound on their $L_2$-norm. Such a result is formulated in
the following

\medskip\noindent
{\bf Lemma 11.3. (Estimate about the $L_2$-norm of the kernel
functions of the $U$-statistics appearing in the diagram formula).}
{\it Let $m$ functions $f_p(x_1,\dots,x_{k_p})$
be given on the products $(X^{k_p},\Cal X^{k_p})$ of some measurable
space $(X,\Cal X)$, $1\le p\le m$, with a probability measure $\mu$
on it, which satisfy inequalities (8.1) and (8.2) (if the index
$k$ is replaced by the index $k_p$ in them), but these
functions need not be canonical. Let us take a coloured diagram
$\gamma\in\Gamma(k_1,\dots,k_m)$, and consider the function
$F_\gamma(f_1,\dots,f_m)$ defined by formulas
(11.9)---(11.12). The $L_2$-norm of the function 
$F_\gamma(f_1,\dots,f_m)$ (with respect to the power of the 
measure~$\mu$ to the space where $F_\gamma(f_1,\dots,f_m)$ is 
defined) satisfies the inequality 
$$
\|F_\gamma(f_1,\dots,f_m)\|_2
\le2^{W(\gamma)}\prod_{p\in U(\gamma)} \|f_p\|_2,
$$ 
where $W(\gamma)$ is given in~(11.13), and the set 
$U(\gamma)\subset\{1,\dots,m\}$ is defined in the following way.
Let us define for a coloured chain  
$\beta=\{(l_1,r_1),(l_2,r_2),\dots,(l_s,r_s)\}\in\gamma$ with 
$1\le l_1<\cdots<l_s\le m$ the set of its interior levels as
and $\text{\rm Int}\,(\beta)=\{l_2,\dots,l_{s-1},l_s\}$ if 
$c_\gamma(\beta)=-1$ and 
$\text{\rm Int}\,(\beta)=\{l_2,\dots,l_{s-1}\}$ if 
$c_\gamma(\beta)=1$. Then we define 
$U(\gamma)=\{1,\dots,m\}\setminus
\(\bigcupp_{\beta\in\gamma}\text{\rm Int}\,(\beta)\)$.}

\medskip
The last result of this section is a corollary of Theorem~11.2. In
this corollary we give an estimate on the expected value of product
of degenerate $U$-statistics. To formulate this result we introduce
the following terminology. Let us call a (coloured) diagram
$\gamma\in\Gamma(k_1,\dots,k_m)$ closed if $c_\gamma(\beta)=1$ for
all chains $\beta\in\gamma$. Let us denote the set of all closed
diagrams by $\bar\Gamma(k_1,\dots,k_m)$. Observe that 
$F_\gamma(f_1,\dots,f_m)$ is constant (a function of zero variable)
for all closed diagram $\gamma\in\bar\Gamma(k_1,\dots,k_m)$, and
$I_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m))
=I_{n,0}(F_\gamma(f_1,\dots,f_m))
=F_\gamma(f_1,\dots,f_m)$
in this case. Now we formulate the following result.

\medskip\noindent
{\bf Corollary of Theorem 11.2 about the expectation of a product
of degenerate $U$-statistics.}
{\it Let a finite sequence of functions $f_p(x_1,\dots,x_{k_p})$,
$1\le p\le m$, be given on the products $(X^{k_p},\Cal X^{k_p})$ of
some measurable space $(X,\Cal X)$ together with a sequence of
independent and identically distributed random variables with 
value in the space $(X,\Cal X)$ which satisfy the conditions
of Theorem 11.2.

Let us apply the notation of Theorem~11.2 together with the notion
of the above introduced class of closed diagrams
$\bar\Gamma(k_1,\dots,k_m)$. The identity
$$
E\(\prod_{p=1}^m k_p! n^{-k_p/2}I_{n,k_p}(f_{k_p})\)
= \sumnl_{\gamma\in\bar\Gamma(k_1,\dots,k_m)} \(\prod_{p=1}^m
J_n(\gamma,p)\) n^{-W(\gamma)/2}\cdot F_\gamma(f_1,\dots,f_m)
\tag11.16
$$
holds. This identity has the consequence
$$
\left|E\(\prod_{p=1}^m k_p! n^{-k_p/2}I_{n,k_p}(f_{k_p})\)\right|
\le \sum_{\gamma\in\bar\Gamma(k_1,\dots,k_m)}
n^{-W(\gamma)/2}|F_\gamma(f_1,\dots,f_m)|.
\tag11.17
$$
Beside this, if $\|f_p\|_2\le\sigma$ for all $1\le p\le m$, then
the numbers $F_\gamma(f_1,\dots,f_m)$ at the right-hand side 
of~(11.17) satisfy the inequality 
$$
|F_\gamma(f_1,\dots,f_m)|\le2^{W(\gamma)}\sigma^{|U(\gamma)|} \quad
\text{for all } \gamma\in\bar\Gamma(k_1,\dots,k_m). \tag11.18
$$
In formula~(11.18) the same number~$W(\gamma)$ and set $U(\gamma)$ 
appear as in Lemma~11.3. The only difference is that in the present 
case $c_\gamma(\beta)=1$ for all chains $\beta\in\gamma$ which 
appear in the definition of $U(\gamma)$.}

\medskip\noindent
{\it Remark:}\/ We have applied a different terminology for 
diagrams in this section and in Section~10, where the theory 
of Wiener--It\^o integrals was discussed. But there is a simple
relation between the terminology of these sections. If we take 
only those diagrams from the diagrams considered in this section 
which contain only chains of length~1 or~2, and beside this the 
chains of length~1 have colour~$-1$, and the chains of length~2 
have colour~1, then we get the diagrams considered in the previous 
section. Moreover, the functions 
$F_\gamma=F_\gamma(f_1,\dots,f_m)$ are the same in the two cases. 
Hence formula~(10.18) in the Corollary of Theorem~10.2 and 
formula~(11.17) in the Corollary of Theorem~11.2 make possible to 
compare the moments of Wiener--It\^o integrals and degenerate 
$U$-statistics. 

The main difference between these estimates is that formula~(11.17) 
contains some additional terms. They are the contributions of 
those diagrams $\gamma\in\bar\Gamma(k_1,\dots,k_m)$ which 
contain chains $\beta\in\gamma$ with length $\ell(\beta)>2$. 
These are those diagrams $\gamma\in\bar\Gamma(k_1,\dots,k_m)$
for which $W(\gamma)>1$. The estimate~(11.18) given for the terms 
$F_\gamma$ corresponding to such diagrams is 
weaker, than the estimate given for the terms $F_\gamma$ with 
$W(\gamma)=0$, since $|U(\gamma)|<m$ if $W(\gamma)\ge1$, while
$|U(\gamma)|=m$, if $W(\gamma)=0$. On the other hand, such terms 
have a coefficient $n^{-W(\gamma)/2}$ at the right-hand side of 
formula~(11.17). A closer study of these formulas may explain the 
relation between the estimates given for the tail distribution 
of Wiener--It\^o integrals and degenerate $U$-statistics.

\beginsection 12. The proof of the diagram formula for $U$-statistics.

In this section the results of the previous section will be proved. 
First I prove its main result, the diagram formula for the product 
of two degenerate $U$-statistics.

\medskip\noindent
{\it Proof of Theorem 11.1.}\/ In the first step of the proof the
product $k_1!I_{n,k_1}(f_1)k_2!I_{n,k_2}(f_2)$ of two degenerate 
$U$-statistics will be rewritten as a sum of not necessarily degenerate 
$U$-statistics. In this step a term by term multiplication is 
carried out for the product $k_1!I_{n,k_1}(f_1)k_2!I_{n,k_2}(f_2)$, 
and the terms of the sum obtained in such a way are put in different 
classes indexed by the (non-coloured) diagrams with two rows of
length~$k_1$ and~$k_2$. This step is very similar to the heuristic 
argument leading to formulas~(10.13) and~(10.13a) in our 
explanation about the diagram formula for Wiener-It\^o integrals.

To carry out this step of the proof consider all sets of pairs
$$
\{(u_1,u'_1),\dots,(u_r,u'_r)\}, \quad 1\le r\le \min(k_1,k_2),
$$
with the following properties: 
$1\le u_1<u_2<\cdots<u_r\le k_1$, the numbers $u'_1,\dots,u'_r$ 
are different, and $1\le u'_s\le k_2$, for all $1\le s\le r$. To a 
set of pairs $\{(u_1,u'_1),\dots,(u_r,u'_r)\}$ with these properties 
let us correspond the following diagram
$\bar\gamma((u_1,u'_1),\dots,(u_r,u'_r))\in\bar\Gamma(k_1,k_2)$, 
where $\bar\Gamma(k_1,k_2)$ denotes the set of (non-coloured) 
diagrams with two rows of length~$k_1$ and~$k_2$. The diagram 
$\bar\gamma((u_1,u'_1),\dots,(u_r,u'_r))$ has two rows, 
$\{1,\dots,k_1\}$, and $\{2,\dots,k_2\}$, its chains of length~2
are the sets $\{(1,u_s),(2,u'_s)\}$, $1\le s\le r$, and beside
this it contains the chains 
$\{(1,r)\}$, $r\in\{1,\dots,k_1\}\setminus\{u_1,\dots,u_r\}$, and 
$\{(2,r)\}$, $r\in\{1,\dots,k_2\}\setminus\{u'_1,\dots,u'_r\}$ of
length~1. All (non-coloured) diagrams 
$\bar\gamma\in\bar\Gamma(k_1,k_2)$ can be represented in the form
$\bar\gamma=\bar\gamma((u_1,u'_1),\dots,(u_r,u'_r))$ with the help 
of a set of pairs $\{(u_1,u'_1),\dots,(u_r,u'_r)\}$, 
$1\le r\le \min(k_1,k_2)$, with the above properties in a unique way.

To make the notation in the subsequent discussion simpler we fix,
similarly to the case of coloured diagrams, an indexation of the
chains of a diagram $\bar\gamma\in\bar\Gamma(k_1,k_2)$, and we
define with its help an indexation of the vertices of this
diagram~$\bar\gamma$, too. Let us take the following natural 
indexation. Consider the diagram $\bar\gamma=
\bar\gamma((u_1,u'_1),\dots,(u_r,u'_r))\in\bar\Gamma(k_1,k_2)$
which has $s(\bar\gamma)=k_1+k_2-r$ chains. The chain 
$\beta\in\bar\gamma$ containing the vertex~$(1,j)$ gets the 
index~$j$, i.e. $(1,j)\in\beta(j)$ for $1\le j\le k_1$. To define 
the index of the remaining chains of~$\bar\gamma$ which are chains
of length~1 of the form $(2,j)$ with 
$j\in\{1,\dots,k_2\}\setminus\{u_1',\dots,u_r'\}$ let us take the 
list $\{\bar l_1,\dots,\bar l_{k_2-r}\}$, 
$1\le\bar l_1<\cdots<\bar l_{k_2-r}$, of the elements of the set 
$\{1,\dots,k_2\}\setminus\{u'_1,\dots,u'_r\}$ in an increasing 
order. Then we define the indices of the remaining chains by the
formula $\beta(k_2+j)=\{(2,\bar l_j)\})$, $1\le j\le k_2-r$.
After this we define the indexation of the vertices of the 
diagram~$\gamma$ by the formula $\alpha_{\bar\gamma}(p,r)=l$ with 
that index~$l$ for which $(p,r)\in\beta(l)$. Let us also 
define the sets 
$V_1=V_1(\bar\gamma)=\{1,\dots,k_1+k_2-r\}\setminus\{u_1,\dots,u_r\}$ 
and $V_2=V_2(\bar\gamma)=\{u_1,\dots,u_r\}$, i.e. $V_1$ is the set 
of indices of the chains of~$\bar\gamma$ of length~1, and $V_2$ is 
the set of indices of the chains of~$\bar\gamma$ of length~2.

Let us consider the product $k_1!I_{n,k_1}(f_1)k_2!I_{n,k_2}(f_2)$, 
and rewrite it in the form of the sum we get by carrying out a term 
by term multiplication in this expression. We put the terms 
obtained in such a way into disjoint classes indexed by the 
 diagrams
$\bar\gamma\in\bar\Gamma(k_1,k_2)$ in the following way: A product
$f_1(\xi_{j_1},\dots,\xi_{j_{k_1}})f_2(\xi_{j'_1},\dots,\xi_{j'_{k_2}})$
belongs to the class indexed by the diagram
$\bar\gamma((u_1,u'_1),\dots,(u_r,u'_r))$ with the parameters 
$(u_1,u'_1),\dots,(u_r,u'_r)$, $1\le r\le \min(k_1,k_2)$, where
$1\le u_1<u_2<\cdots<u_r\le k_1$, the numbers $u'_1,\dots,u'_r$ 
are different, and $1\le u'_s\le k_2$, for all $1\le s\le r$ if the
indices $j_1,\dots,j_{k_1},j'_1,\dots,j'_{k_2}$ in the arguments
of the variables in $f_1(\cdot)$ and $f_2(\cdot)$ satisfy the
relation $j_{u_s}=j'_{u'_s}$, $1\le s\le r$, and there is no more
coincidence between the indices $j_1,\dots,j_{k_1},j'_1,\dots,j'_{k_2}$.

It is not difficult to see by applying the above partition of the terms
in the product $k_1!I_{n,k_1}(f_1)k_2!I_{n,k_2}(f_2)$, and exploiting
that each diagram of $\bar\Gamma(k_1,k_2)$ can be written in the form 
$\bar\gamma((u_1,u'_1),\dots,(u_r,u'_r))$ in a unique way that the
identity
$$
n^{-k_1/2}k_1!I_{n,k_1}(f_1)k_2!n^{-k_2/2}I_{n,k_2}(f_2)
=\sumn_{\bar\gamma\in\bar\Gamma(k_1,k_2)}n^{-(k_1+k_2)/2}s(\bar\gamma)!
I_{n,s(\bar\gamma)}((\overline {f_1\circ f_2})_{\bar\gamma})
\tag12.1
$$
holds, where the functions $(\overline{f_1\circ f_2})_{\bar\gamma}$
are defined in formula (11.3), 
$s(\bar\gamma)=k_1+k_2-|V_2(\bar\gamma)|$ denotes the 
number of chains in $\bar\gamma$, (both chains of length~1 and~2) 
and the notation  $\sum^{\prime(n)}$ means that summation is taken 
only for such  diagrams $\bar\gamma\in\bar\Gamma(k_1,k_2)$ for 
which $n\ge s(\bar\gamma)$. (Let me remark that although 
formula~(11.3) was defined for coloured diagrams, the colours of 
the chains played no role in it.) 

Relation (12.1) is not appropriate for our purposes, since the 
functions $(\overline {f_1\circ f_2})_{\bar\gamma}$ in it may be 
non-canonical. To get the desired formula, Hoeffding's 
decomposition will be applied for the $U$-statistics
$I_{n,s(\bar\gamma)}((\overline {f_1\circ f_2})_{\bar\gamma})$
appearing at the right-hand side of formula~(12.1). This
decomposition becomes slightly simpler because of some special
properties of the function
$(\overline {f_1\circ f_2})_{\bar\gamma}$ related to the
canonical property of the initial functions~$f_1$ and~$f_2$.

To carry out this procedure let us observe that a function
$f(x_{u_1},\dots,x_{u_k})$ is canonical if and only if
$P_{u_s}f(x_{u_1},\dots,x_{u_k})=0$ with the operator $P_{u_s}$
defined in (11.1) for all indices $u_s$, $1\le s\le k$. Beside this, 
the condition that the functions $f_1$ and $f_2$ are canonical 
implies the relation $P_v(\overline {f_1\circ f_2})_{\bar\gamma}=0$ 
for $v \in V_1(\bar\gamma)$ for all diagrams 
$\bar\gamma\in\bar\Gamma(k_1,k_2)$, and this relation remains 
valid if the function $(\overline {f_1\circ f_2})_{\bar\gamma}$ 
is replaced by such functions which we get by applying the 
product of some transforms $P_{v'}$ and $Q_{v'}$, $v'\in P_2$
for the function $(\overline {f_1\circ f_2})_{\bar\gamma}$ with 
the transforms $P$ and $Q$ defined in formulas~(11.1) and~(11.2). 

Beside this, the transforms $P_v$ or $Q_v$ are
exchangeable with the operators $P_{v'}$ or $Q_{v'}$ if
$v\neq v'$, $P_v+Q_v)=I$, where $I$ denotes the
identity operator, and $P_vQ_v=0$, since
$P_vQ_v=P_v-P^2_v=0$. The above relations
make possible the following decomposition of the function
$(\overline {f_1\circ f_2})_{\bar\gamma}$ for all
$\bar\gamma\in\bar\Gamma(k_1,k_2)$ to the sum of canonical
functions (just as it was done in the Hoeffding decomposition): 
$$ \allowdisplaybreaks
\align
(\overline {f_1\circ f_2})_{\bar\gamma}
&=\prod_{v\in V_2}
(P_v+Q_v)(\overline {f_1\circ f_2})_{\bar\gamma}\\
&=\sum_{A\subset V_2}\(\prod_{v\in A} P_v
\prod_{v\in V_2\setminus A}Q_v\)
(\overline {f_1\circ f_2})_{\bar\gamma}
=\sum_{\gamma\in\Gamma(\bar\gamma)}(f_1\circ f_2)_\gamma, \tag12.2 
\endalign 
$$
where the function $(f_1\circ f_2)_\gamma$ is defined in 
formula~(11.4), and $\Gamma(\bar\gamma)$ denotes the set of those
coloured diagrams $\gamma\in\Gamma(k_1,k_2)$ which consist of 
those chains (with a colour $\pm1$) as the non-coloured 
diagram~$\bar\gamma$. (Clearly, $s(\gamma)=s(\bar\gamma)$ for the 
number of chains of~$\gamma$ and~$\bar\gamma$ if
$\gamma\in\Gamma(\bar\gamma)$.) Indeed, given a set $A\subset V_2$, 
we have $(\prodd_{v\in A}P_v \prodd_{v\in V_2\setminus A} Q_v)
(\overline {f_1\circ f_2})_{\bar\gamma}=(f_1\circ f_2)_\gamma$ 
with that coloured diagram $\gamma\in\Gamma(\bar\gamma)$ whose
chains with colour~$1$ are the chains $\beta(l)\in\bar\gamma$
with $l\in A$, and which contains the remaining chains 
$\beta(l)\in\bar\gamma$ with colour~$-1$. Then we get 
relation~(12.2) by summing up this identity for 
all~$A\subset V_2$. The function $(f_1\circ f_2)_\gamma$ 
corresponding to the coloured diagram obtained with the help of 
the set~$A$ has $|O(\gamma)|=k_1+k_2-|V_2(\bar\gamma)|-|A|$ 
variables, where $|O(\gamma)|$ is the number of open indices 
in~$\gamma$.
 
Let us consider the functions $F_\gamma(f_1,f_2)$, 
$\gamma\in\Gamma(k_1,k_2)$, defined in~(11.5) which means a reindexation
of the functions $(f_1\circ f_2)_\gamma$ to get functions with
variables $x_1$,\dots, $x_{|O(\gamma)|}$. We claim that 
$$
\aligned
&n^{-(k_1+k_2)/2}|O(\bar\gamma)|! I_{n,\bar s(\bar\gamma)}\((\overline
{f_1\circ f_2})_{\bar\gamma}\) \\
&\qquad =\sum_{\gamma\in\Gamma(\bar\gamma)}n^{-(k_1+k_2)/2}  
n^{|C(\gamma)|}J_n(\gamma) |O(\gamma)|! 
I_{n,|O(\gamma)|}\(F_\gamma(f_1,f_2)\)
\endaligned \tag12.3
$$
with $J_n(\gamma)=1$ if $|C(\gamma)|=0$, and
$$
J_n(\gamma)=\prodd_{j=1}^{|C(\gamma)|}\(\frac{n-s(\gamma)+j}n\)
\quad \text{if } \; |C(\gamma)|>0. \tag12.4
$$
for all $\bar\gamma\in\bar\Gamma(k_1,k_2)$.

Since $I_{n,|O(\gamma)|}\(F_\gamma(f_1,f_2)\)
=I_{n,|O(\gamma)|}\(f_1\circ f_2)_\gamma\)$ relation~(12.3) follows
from relation~(12.2) just as formula~(9.3) follows from formula~(9.2) 
in the proof of the Hoeffding decomposition. Let us understand why 
the coefficient $n^{|C(\gamma)|}J_n(\gamma)$ appears at the right-hand 
side of~(12.3).

This coefficient can be calculated in the following way. Take a 
general term 
$(f_1\circ f_2)_\gamma(\xi_{j_{l_u}},\,l_u\in O(\gamma))$ in the
$U$-statistic $|O(\gamma)|!I_{n,|O(\gamma)|}((f_1\circ f_2)_\gamma)$,
and calculate the number of terms 
$(\overline{f_1\circ f_2})_{\bar\gamma}
(\xi_{j'_1},\xi_{j'_2}\dots,\xi_{j'_{s(\bar\gamma)}})$ in the
$U$-statistic  $|O(\bar\gamma)|! I_{n,\bar s(\bar\gamma)}
((\overline{f_1\circ f_2})_{\bar\gamma})$ for which the sequence 
of indices $(j'_1,\dots,j'_{s(\bar\gamma)})$ satisfies the relation
$j'_{l_u}=j_{l_u}$ for all $l_u\in O(\gamma)$. I claim that it 
equals $n^{|C(\gamma)|}J_n(\gamma)$. It can be seen that 
this number $n^{|C(\gamma)|}J_n(\gamma)$ appears as the coefficient
at right-hand side of~(12.3).

Indeed, we have to calculate the number of such sequences 
$j'_1,j'_2,\dots,j'_{s(\bar\gamma)}$ for which the value 
$j'_{l_u}=j_{l_u}$ is prescribed for the indices 
$l_u\in O(\gamma)$, and the other elements of the 
sequence can take arbitrary integer value between~1 and~$n$ 
with the only restriction that all elements of the sequence 
$j'_1,j'_2,\dots,j'_{s(\bar\gamma)}$ must be different. 
The number of such sequences equals 
$(n-|O(\gamma)|)(n-|O(\gamma)|-1)\cdots(n-|C(\gamma)|-|O(\gamma)|+1)
=J_n(\gamma)n^{|C(\gamma)|}$. (In this calculation we exploited the
fact that $|O(\gamma)|+|C(\gamma)|=s(\gamma)$.)

Let us observe that $k_1+k_2-2|C(\gamma)|=|O(\gamma)|+W(\gamma)$
with the number $W(\gamma)$ introduced in the formulation of
Theorem~11.1. Hence 
$$
n^{-(k_1+k_2)/2}n^{|C(\gamma)|}=n^{-W(\gamma)/2}n^{-|O(\gamma)|/2}.
$$ 
Let us replace the left-hand side of the last identity by its 
right-hand side in~(12.3), and let us sum up the identity
we get in such a way for all $\bar\gamma\in\bar\Gamma(k_1,k_2)$ 
such that $s(\bar\gamma)\le n$. The identity we get in 
such a way together with formulas~(12.1) and~(12.4) imply the 
identity~(11.6). Clearly, $I_{n,|O(\gamma)|}(F_\gamma(f_1,f_2))=
I_{n,|O(\gamma)|}(\text{\rm Sym}F_\gamma(f_1,f_2))$, hence the 
term $I_{n,|O(\gamma)|}(F_\gamma(f_1,f_2))$ can be replaced by
$I_{n,|O(\gamma)|}(\text{\rm Sym}F_\gamma(f_1,f_2))$ in
formula~(11.6). We still have to prove inequalities~(11.7) 
and~(11.8).

\medskip
Inequality (11.7), the estimate of the $L_2$-norm of the function
$(f_1\circ f_2)_\gamma$ follows from the Schwarz inequality, and
actually it agrees with inequality (10.11), proved at the start
of Appendix~B. Hence its proof is omitted here.

To prove inequality (11.8) let us introduce, similarly to
formula (11.2), the operators
$$
\tilde Q_{u_j}h(x_{u_1},\dots,x_{u_r})=h(x_{u_1},\dots,x_{u_r})+
\int h(x_{u_1},\dots,x_{u_r})\mu(\,dx_{u_j}), \quad 1\le j\le r,
\tag 12.5
$$
in the space of functions $h(x_{u_1},\dots,x_{u_r})$ with coordinates
in the space $(X,\Cal X)$. (The indices $u_1,\dots,u_r$ are all
different.) Observe that both the operators $\tilde Q_{u_j}$ and the
operators $P_{u_j}$ defined in (11.1) are positive, i.e. these
operators map a non-negative function to a non-negative function.
Beside this, $Q_{u_j}\le\tilde Q_{u_j}$, and the norms of the
operators  $\frac{\tilde Q_{u_j}}2$ and $P_{u_j}$ are bounded by 1
both in the $L_1(\mu)$, the $L_2(\mu)$ and the supremum norm.

Let us define the function
$$
(\widetilde{f_1\circ f_2})_\gamma(x_j,\;j\in O(\gamma))
=\(\prod_{j\in C(\gamma)}P_j \prod_{j\in O_2(\gamma) } \tilde Q_j\) 
\overline{(f_1\circ f_2)}_\gamma(x_j,\;j\in C(\gamma)\cup O(\gamma))
\tag12.6
$$
with the notation of Section~11. The function
$(\widetilde{f_1\circ f_2})_\gamma$ was defined with the help of
$\overline{(f_1\circ f_2)}_\gamma$ similarly to
$(f_1\circ f_2)_\gamma$ defined in~(11.4), only  the operators $Q_j$
were replaced by $\tilde Q_j$ in its definition.

In the proof of (11.8) it may be assumed that $\|f_1\|_2\le\|f_2\|_2$.
The properties of the operators $P_{u_j}$ and $\tilde Q_{u_j}$
listed above together with the condition
$\sup|f_2(x_1,\dots,x_k)|\le1$ imply that
$$
|(f_1\circ f_2)_\gamma|\le (\widetilde{|f_1|\circ |f_2|})_\gamma\le
(\widetilde{|f_1|\circ 1})_\gamma, \tag12.7
$$
where `$\le$' means that the function at the right-hand side is
greater than or equal to  the function at the left-hand side in all
points, and the term~1 in~(12.7) denotes the function which equals 
identically~1. Because of the identity 
$\|F_\gamma(f_1,f_2)\|_2=\|(f_1\circ f_2)_\gamma\|_2$ and 
relation~(12.7) it is enough to show that
$$
\aligned
\|(\widetilde{|f_1|\circ1})_\gamma\|_2&=
\left\|\(\prod_{j\in C(\gamma)}  P_j
\prod_{j\in O_2(\gamma)}  \tilde Q_j\)
|f_1(x_{\alpha_\gamma(1,1)},\dots,x_{\alpha_\gamma(1,k_1)})|\right\|_2 \\
&\le 2^{W(\gamma)}\|f_1\|_2
\endaligned \tag12.8
$$
to prove relation~(11.8). But this inequality trivially holds, since
the norm of all operators $P_j$ in formula (12.8) is bounded
by~1, the norm of all operators $\tilde Q_j$ is bounded
by~2 in the $L_2(\mu)$ norm, and $|O_2(\gamma)|=W(\gamma)$.

\medskip\noindent
{\it Proof of Theorem 11.2.} Theorem~11.2 will be proved with the
help of Theorem~11.1 by induction with respect to the number of
degenerate $U$-statistics $k_p!I_{n,k_p}(f_p)$, $1\le p\le m$. 
Formula~(11.15) holds for $m=2$ by Theorem~11.1. To prove it for 
a general parameter~$m$ let us first fix a coloured diagram 
$\bar\gamma\in\Gamma(k_1,\dots,k_{m-1})$ and consider the set of
diagrams of~$m$ rows which are its `continuation', i.e. let
$$\
\Gamma(\bar\gamma)=\{\gamma\colon\; \gamma\in\Gamma(k_1,\dots,k_m),\,
\gamma_{pr}=\bar\gamma\}.
$$ 
(Here we work with the diagrams $\gamma_{pr}$ and $\gamma_{cl}$
introduced for a diagram~$\gamma\in\Gamma(k_1,\dots,k_m)$ in the
previous section.) I claim that
$$
\aligned
&n^{-|O(\bar\gamma)|/2}|O(\bar\gamma)|!I_{n,|O(\bar\gamma)|}
(F_{\bar\gamma}(f_1,\dots,f_{m-1}))
\cdot n^{-k_m/2}k_m!I_{n,k_m}(f_m)\\
&\qquad=\sumn_{\gamma\in\Gamma(\bar\gamma)}
\prod_{j=1}^{|C(\gamma_{cl})|}\(\frac{n-s(\gamma_{cl})+j}n\) 
n^{-W(\gamma_{cl})/2} \\
&\qquad\qquad\qquad n^{-|O(\gamma)|/2}|O(\gamma))|!
I_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m)),
\endaligned\tag12.9
$$
where $\sumn$ means that summation is taken for such
$\gamma\in\Gamma(\bar\gamma)$ for which $s(\gamma_{cl})\le n$, and
$\prodd_{j=1}^{|C(\gamma_{cl})|}$ equals~1, if $|C(\gamma_{cl})|=0$.

Relation~(12.9) can be checked by applying Theorem~11.1 for 
the pair of functions $F_{\bar\gamma}(f_1,\dots,f_{m-1})$ 
and~$f_m$. To get it first we show that there is a mutual 
correspondence between the coloured diagrams
$\gamma\in\Gamma(|O(\bar\gamma)|,k_m)$ and the class of 
diagrams $\{\gamma_{cl}\colon\;\gamma\in\Gamma(\bar\gamma)\}$ 
in such a way that if $\gamma\in\Gamma(\bar\gamma)$ and 
$\gamma'\in\Gamma(|O(\gamma)|,k_m)$ correspond to each other, 
then $\gamma'=\gamma_{cl}$. We shall fix an enumeration of 
the chains of the diagram~$\bar\gamma$, and we shall make 
such an enumeration of the chains of a diagram 
$\gamma\in\Gamma(\bar\gamma)$ for which the enumeration of 
the chains of $\bar\gamma$ and $\gamma_{pr}$ agree. The 
correspondence between the above mentioned two classes of 
diagrams depends on the enumeration of the chains 
of~$\bar\gamma$, but this will cause no problem. To get it 
observe that for each $\gamma\in\Gamma(\bar\gamma)$ there is 
a diagram $\gamma'=\gamma_{cl}\in\Gamma(|O(\bar\gamma)|,k_m)$. 
On the other hand, I claim that for all diagrams 
$\gamma'\in \Gamma(|O(\bar\gamma)|,k_m)$ such a diagram 
$\gamma(\gamma')\in\Gamma(\bar\gamma)$ can be found for which
$\gamma(\gamma')_{cl}=\gamma'$. 

This diagram $\gamma(\gamma')\in\Gamma(\bar\gamma)$ will be 
defined in the following way. Let 
$\bar l_1,\bar l_2,\dots,\bar l_{|O(\bar\gamma)|}$ be the 
indices of the chains of the diagram~$\bar\gamma$ with colour~$-1$.
The diagram $\gamma(\gamma')$ will be defined so that the chains 
of colour~$1$ of $\bar\gamma$ will be chains of colour~1 of 
$\gamma(\gamma')$, too. If the vertex $(1,j)$ of the 
diagram~$\gamma'$ is contained in a chain of length~1, then the 
diagram~$\gamma(\gamma')$ contains the chain $\beta(\bar l_j)$
with colour~$-1$. If this vertex is contained in a chain 
$\{(1,j),(2,r_j)\}\in\gamma'$ of length~2, then $\gamma(\gamma')$ 
contains the diagram $\beta(\bar l_j)\cup\{(m,r_j)\}$ with the
same colour as the chain $\{(1,j),(2,r_j)\}$ has in $\gamma'$.
Finally, if the vertex $(2,r)$  is contained in the chain $\{(2,r)\}$ 
of length~1 in $\gamma'$, then  $\{(m,r)\}$ will be a chain of 
length~1 of~$\gamma(\gamma')$  with colour~$-1$. In such a way we 
get such a diagram $\gamma(\gamma')\in\Gamma(\bar\gamma)$ for which 
$\gamma(\gamma')_{cl}=\gamma'$. 
%(The above construction of the set $\gamma(\gamma')$ depends on the 
%enumeration of the chains of the diagram~$\bar\gamma$, but we get 
%the same class of diagrams for two different enumerations 
%of~$\bar\gamma$ if we take the diagrams $\gamma(\gamma')$ for all 
%$\gamma'\in\Gamma(|O(\bar\gamma)|,k_m)$.)

We get relation~(12.9) by applying Theorem~11.1 for the product
$$
n^{-|O(\bar\gamma)|/2}|O(\bar\gamma)|!I_{n,|O(\bar\gamma)|}
(F_{\bar\gamma}(f_1,\dots,f_{m-1}))\cdot n^{-k_m/2}k_m!I_{n,k_m}(f_m)
$$
and writing all diagrams~$\gamma'\in\Gamma(|O(\gamma)|,k_m)$
in the form $\gamma_{cl}$, where $\gamma_{cl}$ is the
closing diagram of the diagram
$\gamma(\gamma')\in\Gamma(\bar\gamma)$ defined in the previous
paragraph.

Relation~(11.15) for the parameter~$m$ can be proved with the help
of relation~(12.9) and the inductive assumption by which it holds 
for~$m-1$. Indeed, let us multiply formula~(12.9) by 
$\prodd_{p=2}^{m-1}J_n(\bar\gamma,p)n^{-W(\bar\gamma)/2}$, and sum 
up this identity for all such diagrams 
$\bar\gamma\in\Gamma(k_1,\dots,k_{m-1})$ for which 
$B_1(\gamma,p)+B_2(\bar\gamma,p)\le n$ for all $2\le p\le m-1$. 
Then the sum of  the terms at the left-hand side equals the 
left-hand side of formula~(11.15) for parameter~$m$. 

I claim that the sum of the terms at the right-hand side equals 
the right-hand side of formula~(11.15) for parameter~$m$. To see 
this it is enough to check that for all 
$\gamma\in\Gamma(\bar\gamma)$ we have $W(\bar\gamma)+W(\gamma_{cl})
=W(\gamma_{pr})+W(\gamma_{cl})=W(\gamma)$,
$\prodd_{p=2}^{m-1} J_n(\gamma_{pr},p)
\prodd_{j=1}^{|C(\gamma_{cl})|}\(\frac{n-s(\gamma_{cl})+j}n\)
=\prodd_{p=2}^m J_n(\gamma,p)$, where 
$\prodd_{j=1}^{|C(\gamma_{cl})|}=1$ if $|C(\gamma_{cl})|=0$, 
and the relation $B_1(\gamma,p)+B_2(\gamma,p)\le n$ holds for all 
$2\le p\le m$ if and only if 
$B_1(\gamma_{pr},p)+B_2(\gamma_{pr},p)\le n$ 
for all $2\le p\le m-1$, and $s(\gamma_{cl})\le n$. But these 
relations can be simply checked. The identity about the 
function~$W(\cdot)$ can be checked by taking into account the
definition of the diagrams~$\gamma_{pr}$ and~$\gamma_{cl}$, in
particular the colouring of the chains in these diagrams. The 
remaining relations can be proved with the help of the 
observation that for a diagram $\gamma\in\Gamma(k_1,\dots,k_m)$ 
$B_1(\gamma_{pr},p)=B_1(\gamma,p)$ and
$B_2(\gamma_{pr},p)=B_2(\gamma,p)$ for all $2\le p\le m-1$.
Beside this $|C(\gamma_{cl})|=B_1(\gamma,m)$ and
$|O(\gamma_{cl})|=B_2(\gamma,m)$. Theorem~11.2 is proved.

\medskip\noindent
{\it Proof of Lemma 11.3.} The proof is similar to that of
formula (11.8) at the end of Theorem~11.1. Let us define the
functions $\tilde F_{\gamma}(f_1,\dots,f_p)$, 
$\gamma\in\Gamma(k_1,\dots,k_p)$, recursively for all 
$2\le p\le m$ similarly to the definition of the functions 
$F_\gamma(f_1,\dots,f_p)$ with the difference that
the operator $Q_{u_j}=I-P_{u_j}$ is replaced by 
$\tilde Q_{u_j}=I+P_{u_j}$ in the new definition. Then we have 
$|F_\gamma(f_1,\dots,f_m)|\le\tilde F_\gamma(|f_1|,\dots,|f_m|)$ 
in all points. Hence $\|F_\gamma(f_1,\dots,f_m)\|_2\le
\|\tilde F_\gamma(f_1,\dots,f_m)\|_2$, and to prove Lemma~11.3 
it is enough to show that
$$
\|\tilde F_\gamma(|f_1|,\dots,|f_m|)\|_2\le 2^{W(\gamma)}
\prod_{p\in U(\gamma)}\|f_p\|_2 \quad\text{if } 
\gamma\in\Gamma(k_1,\dots,k_m)  \tag12.10
$$
with the same number $W(\gamma)$ and set $U(\gamma)$ which 
were considered in Lemma~11.3. Relation~(12.10) will be proved by
induction with respect to~$m$.

Relation~(12.10) holds for $m=2$. Indeed, if $W(\gamma)=0$, then
$U(\gamma)=\{1,2\}$, we have $\tilde F_\gamma=F_\gamma$, and 
formula~(11.7) supplies the estimate. If $W(\gamma)\ge1$, then 
$U(\gamma)=\{1\}$, and actually in the proof of relation~(11.8) 
we proved this relation.

In the case $m>2$ this inequality will be proved by induction with
the help of the identity (with the notation of formula~(11.3)
$$
\aligned
&\|\tilde F_\gamma(|f_1|,\dots,|f_m|)\|_2
=\biggl\|\(\prod_{p\in C(\gamma_{cl})} 
P_{p}\prod_{p\in O_2(\gamma_{cl})} \tilde Q_p\)\\ 
&\qquad\qquad\qquad \overline{(\tilde F_{\gamma_{pr}}(|f_1|,\dots,
|f_{m-1}|)\circ |f_m|)}_{\gamma_{cl}}
(x_p,\;p\in O(\gamma_{cl})\cup C(\gamma_{cl})) \biggr\|_2.
\endaligned \tag12.11
$$
In the case $W(\gamma_{cl})=0$, i.e. if $\gamma_{cl}$
contains no open chain of length~2 we have 
$U(\gamma)=U(\gamma_{pr})\cup\{m\}$, $W(\gamma)=W(\gamma_{pr})$, 
and formula~(2.11) contains no operator~$\tilde Q_p$. In this 
case inequality~(12.10) follows from the representation of 
$\|\tilde F_\gamma(|f_1|,\dots,|f_m|)\|_2$ given in~(12.11), 
relation~(11.7) and from the inductive hypothesis by which 
inequality~(12.10) holds for
$\|(\tilde F_{\gamma_{pr}}(|f_1|,\dots,|f_{m-1}|)\|_2$.

In the case $W(\gamma_{cl})>0$ we have $U(\gamma)=U(\gamma_{pr})$, 
$W(\gamma)=W(\gamma_{pr})+W(\gamma_{cl})$, and inequality~(12.10)
can be proved similarly to the case $W(\gamma_{cl})=0$ with the
only difference that in this case instead of~(11.7) we have to apply  
that strengthened version of~(11.8) which is contained in 
formula~(12.10) in the special case $m=2$. Lemma~11.3 is proved.

\medskip
The corollary of Theorem 11.2 is a simple consequence of
Theorem~11.2 and Lem\-ma~11.3.

\medskip\noindent
{\it Proof of the corollary of Theorem 11.2.}\/ Observe that
$F_\gamma$ is a function of $|O(\gamma)|$ arguments. Hence a 
coloured diagram $\gamma\in\Gamma(k_1,\dots,k_m)$ is in the 
class of closed diagrams, i.e. 
$\gamma\in\bar\Gamma(k_1,\dots,k_m)$ if and only if
$F_\gamma(f_1,\dots,f_m)$ is a constant. Thus formula~(11.16) 
is a simple consequence of relation (11.15) and the observation 
that $EI_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m))=0$ if 
$|O(\gamma)|\ge1$, i.e. if
$\gamma\notin\bar\Gamma(k_1,\dots,k_m)$, and
$$
\align
I_{n,|O(\gamma)|}(F_\gamma(f_1,\dots,f_m))&=
I_{n,0}(F_\gamma(f_1,\dots,f_m))=F_\gamma(f_1,\dots,f_m) \\
&\qquad\qquad \text{ if } \gamma\in\bar\Gamma(k_1,\dots,k_m).
\endalign
$$ 
Relations~(11.17) and~(11.18) follow from relation~(11.16) and 
Lemma~11.3.

\beginsection 13. The proof of Theorems 8.3, 8.5 and Example 8.7.

This section contains the proof of the estimates on the distribution
of a multiple Wiener--It\^o integral or degenerate $U$-statistic
formulated in Theorems~8.5 and~8.3 together with the proof of
Example~8.7. Beside this, also a multivariate version of Hoeffding's
inequality (Theorem~3.4) will be proved here. The latter result
is useful in the estimation of the supremum of degenerate
$U$-statistics. The estimate on the distribution of a multiple
random integral with respect to a normalized empirical
distribution given in Theorem~8.1 is omitted, because, as it was
shown in Section~9, this result follows from the estimate of
Theorem~8.3 on degenerate $U$-statistics. This section will be
finished with a separate part Section~13~B, where the results
proved in this section are discussed together with the method of
their proofs and some recent results.

The proof of Theorems~8.5 and~8.3 is based on a good estimate on
high moments of Wiener--It\^o integrals and degenerate
$U$-statistics. These estimates follow from the corollaries of 
Theorems~10.2 and~11.2. Such an approach slightly differs from the
classical proof in the one-variate case. The natural one-variate
version of the problems discussed here is an estimate about the
tail distribution of a sum of independent random variables. This 
estimate is generally proved with the help of a good bound on 
the moment generating function of the sum. Such a method may
not work in the multivariate case, because, as later 
calculations will show, there is no good estimate on the 
moment-generating function estimate of $U$-statistics or multiple 
Wiener--It\^o integrals of order $k\ge3$. Actually, the 
moment-generating function of a Wiener--It\^o integral of order 
$k\ge3$ is always divergent, because the tail behaviour of such 
a random integral is similar to that of the $k$-th power of a 
Gaussian random variable. On the other hand, good bounds on the 
moments $EZ^{2M}$ of a random variable~$Z$ for all positive 
integers~$M$ (or at least for a sufficiently rich class of 
parameters~$M$) together with the application of the Markov 
inequality for $Z^{2M}$ and an appropriate choice of the 
parameter~$M$ yield a good estimate on the distribution of~$Z$.

Propositions~13.1 and~13.2 give estimates on the moments of
Wiener--It\^o integrals and degenerate $U$-statistics.

\medskip\noindent
{\bf Proposition 13.1. (Estimate of the moments of Wiener--It\^o
integrals).} {\it Let $f(x_1,\dots,x_k)$ be a function of $k$
variables on some measurable space $(X,\Cal X)$ that satisfies
formula (8.12) with some $\sigma$-finite measure $\mu$. Take
the $k$-fold Wiener--It\^o integral $Z_{\mu,k}(f)$ of this function
with respect to a white noise $\mu_W$ with reference measure~$\mu$.
The inequality
$$
E\(k!|Z_{\mu,k}(f)|\)^{2M}\le 1\cdot3\cdot5\cdots
(2kM-1)\sigma^{2M}\quad\text {for all }M=1,2,\dots       \tag13.1
$$
holds.}

\medskip
By Stirling's formula Proposition~13.1 implies that
$$
E(k!|Z_{\mu,k}(f)|)^{2M}\le \frac{(2kM)!}{2^{kM}(kM)!}\sigma^{2M}
\le A\(\frac2e\)^{kM}(kM)^{kM}\sigma^{2M} \tag13.2
$$
for any $A>\sqrt2$ if $M\ge M_0=M_0(A)$. Formula (13.2) can be
considered as a simpler, better applicable version of
Proposition~13.1. It can be better compared with the moment estimate
on~degenerate $U$-statistics given in~(13.3).

Proposition~13.2 provides a similar, but weaker inequality for the
moments of normalized degenerate $U$-statistics.

\medskip\noindent
{\bf Proposition 13.2. (Estimate on the moments of degenerate
$U$-statistics).} {\it Let us consider a degenerate $U$-statistic
$I_{n,k}(f)$ of order $k$ with sample size $n$ and with a kernel
function $f$ satisfying relations (8.1) and (8.2) with some
$0<\sigma^2\le1$. Fix a positive number $\eta>0$. There exist some
universal constants $A=A(k)>\sqrt2$, $C=C(k)>0$ and $M_0=M_0(k)\ge1$
depending only on the order of the $U$-statistic $I_{n,k}(f)$ such
that
$$
\aligned
E\(n^{-k/2}k!I_{n,k}(f)\)^{2M}&\le A\(1+C\sqrt\eta\)^{2kM}
\(\frac2e\)^{kM}\(kM\)^{kM}\sigma^{2M} \\
&\qquad \text{for all integers } M \text{ such that } kM_0\le kM\le
\eta n\sigma^2.
\endaligned \tag13.3
$$

In formula (13.3) such a constant $C=C(k)$ can be chosen which does 
not depend on the order $k$ of the $U$-statistic $I_{n,k}(f)$.
For instance $C=4$ is an appropriate choice.} 

\medskip
Theorem~13.2 yields a good estimate on
$E\(n^{-k/2}k!I_{n,k}(f)\)^{2M}$ with a fixed exponent $2M$ with
the choice $\eta=\frac{kM}{n\sigma^2}$. With such a choice of the
number $\eta$ formula~(13.3) yields an estimate on the moments
$E\(n^{-k/2}k!I_{n,k}(f)\)^{2M}$ comparable with the estimate on
the corresponding Wiener--It\^o integral if $M\le n\sigma^2$, while
it yields a much weaker estimate if $M\gg n\sigma^2$.

Now I turn to the proof of these propositions.

\medskip\noindent
{\it Proof of Proposition 13.1.}\/ Proposition 13.1 can be simply
proved by means of the Corollary of Theorem 10.2 with the choice
$m=2M$, and $f_p=f$ for all $1\le p\le 2M$. Formulas~(10.18)
and~(10.19) yield that
$$
E\(k!Z_{\mu,k}(f)^{2M}\)\le\( \int
f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(dx_k)\)^M|\Gamma_{2M}(k)|\le
|\Gamma_{2M}(k)|\sigma^{2M},
$$
where $|\Gamma_{2M}(k)|$ denotes the number of closed diagrams
$\gamma$ in the class
$\bar\Gamma(\underbrace{k,\dots,k}_{2M\text{ times}})$
introduced in the corollary of Theorem~10.2. Thus to complete the 
proof of Proposition~13.1 it is enough to show that
$|\Gamma_{2M}(k)|\le 1\cdot3\cdot5\cdots(2kM-1)$. But this can
easily be seen with the help of the following observation. Let
$\bar\Gamma_{2M}(k)$ denote the class of all graphs with vertices
$(l,j)$, $1\le l\le 2M$, $1\le j\le k$, such that from all vertices
$(l,j)$ exactly one edge starts, all edges connect different
vertices, but edges connecting vertices $(l,j)$ and $(l,j')$ with
the same first coordinate~$l$ are also allowed. Let
$|\bar\Gamma_{2M}(k)|$ denote the number of graphs in
$\bar\Gamma_{2M}(k)$. Then clearly
$|\Gamma_{2M}(k)|\le|\bar\Gamma_{2M}(k)|$. On the other hand,
$|\bar\Gamma_{2M}(k)|=1\cdot3\cdot5\cdots(2kM-1)$. Indeed, let us
list the vertices of the graphs from $\bar\Gamma_{2M}(k)$ in an
arbitrary way. Then the first vertex can be paired with another
vertex in $2kM-1$ way, after this the first vertex from which no
edge starts can be paired with $2kM-3$ vertices from which no edge
starts. By following this procedure the next edge can be chosen
$2kM-5$ ways, and by continuing this calculation we get the desired
formula.

\medskip\noindent
{\it Proof of Proposition 13.2.}\/ Relation (13.3) will be proved by
means of relations (11.17) and (11.18) in the Corollary of
Theorem~11.2 with the choice $m=2M$ and $f_p=f$ for all
$1\le p\le 2M$. Let us take the class of closed coloured diagrams
$\Gamma(k,M)=\bar\Gamma(\underbrace{k,\dots,k}_{2M\text{times}})$.
This will be partitioned into subclasses 
$\Gamma(k,M,r)$, $1\le r\le kM$, where $\Gamma(M,k,r)$ contains 
those closed diagrams $\gamma\in\Gamma(k,M)$ for which 
$W(\gamma)=2r$. Let us recall that $W(\gamma)$ was defined 
in~(11.13), and in the case of closed diagrams 
$W(\gamma)=\summ_{\beta\in\gamma}(\ell(\beta)-2)$. For a diagram 
$\gamma\in\Gamma(k,M)$, $W(\gamma)$ is an even number, since 
$W(\gamma)+2s(\gamma)=2kM$, where $s(\gamma)$ denotes the 
number of chains in~$\gamma$.

First we prove an estimate about the cardinality of~$\Gamma(M,k,r)$.
We claim that there exist some constant $A=A(k)>0$ and threshold 
index $M_0=M_0(k)$ depending only the order~$k$ of the~$U$-statistic
$I{n,k}(f)$ for which
$$
|\Gamma(k,M,r)|\le A\binom{2kM}{2r}\(\frac2e\)^{kM}(kM)^{kM+r}2^{2r}
\quad\text{for all }0\le r\le kM  \tag13.4
$$
if $A\ge A_0(k)$ and $M\ge M_0(k)$.

To prove formula~(13.4) we define a map $T\colon\;\gamma\to T(\gamma)$
from the set of diagrams $\gamma\in\Gamma(k,M,r)$ to the set of 
paired diagrams in such a way that $T(\gamma)\neq T(\gamma')$ if 
$\gamma\neq\gamma'$, and give a good bound on the number of 
paired diagrams~$T(\gamma)$, $\gamma\in\Gamma(k,M,r)$, obtained
in such a way. (We shall call a diagram $\gamma$ a paired  diagram, 
if all of its chains have length~2, i.e. the have the form
$\beta=\{(p,r),(p',r')\}\in\gamma$, with~$p\neq p'$.) To define the 
paired diagrams we shall work with first we introduce the set 
$\Cal W(\gamma)=\bigcupp_{\beta\in\gamma}
\{(p_2(\beta),q_2(\beta)),\dots,(p_{s-1}(\beta),q_{s-1}(\beta))\}$,
for all $\gamma\in \Gamma(k,M,r)$, where 
$\beta=\{(p_1(\beta),q_1(\beta)),\dots,(p_s(\beta),q_s(\beta))\}$
with $1\le p_1(\beta)<p_2(\beta)<\cdots<p_s(\beta)\le 2M$ for all
$\beta\in\gamma$, i.e. $\Cal W(\gamma)$ is the set of vertices we
get by omitting the first and last vertices of all chains 
$\beta\in\gamma$, and then taking the union of the vertices of 
these diminished chains. Observe that $|\Cal W(\gamma)|=W(\gamma)$ 
for a closed diagram.

We take a copy $(p,q,C)$ of all elements $(p,q)\in\Cal W(\gamma)$ 
of a diagram~$\gamma\in\Gamma(k,M,r)$. First we define the set of 
vertices $V(T(\gamma))$ of the paired diagram $T(\gamma)$. It is 
a set of vertices consisting of $2M$ rows, and its $p$-th row is 
$\{(p,1),\dots,(p,k_p)\}\cup\{(p,q,C)\colon\;\,(p,q)\in\Cal W(\gamma)\}$
for all $1\le p\le 2M$. We have 
$|V(T(\gamma)|=2kM+|\Cal W(\gamma)|=2kM+2r$. We define the paired 
diagram $T(\gamma)$ on the set $V(T(\gamma))$ in the following way.
Given a chain
$\beta=\{(p_1(\beta),q_1(\beta)),\dots,(p_s(\beta),q_s(\beta))\}\in\gamma$,
with $1\le p_1(\beta)<p_2(\beta)<\cdots<p_s(\beta)\le 2M$, we 
correspond to it the following sets of pairs (chains of length~2) 
in $V(T(\gamma))$: 
$$
\align
&\{((p_1(\beta),q_1(\beta)),((p_2(\beta),q_2(\beta),C)\},
\{((p_2(\beta),q_2(\beta)),((p_3(\beta),q_3(\beta),C)\},\dots, \\
&\qquad \{((p_{s-2}(\beta),q_{s-2}(\beta)),
((p_{s-1}(\beta),q_{s-1}(\beta),C)\},
\{((p_{s-1}(\beta),q_{s-1}(\beta)),((p_s(\beta),q_s(\beta)\}.
\endalign
$$
(In the case $\ell(\beta)=2$, we map~$\beta$ to itself.)
Defining these pairs of vertices for all $\beta\in\gamma$ we get
the paired diagram~$T(\gamma)$ with the desired properties.

The number of the above defined sets~$V(T(\gamma))$, 
$\gamma\in\Gamma(k,M,r)$, is less than or equal to
$\binom{2kM}{2r}$, and each of these sets $V(T(\gamma))$ has 
$2kM+2r$ vertices. Hence the number of paired diagrams with 
vertices in a fixed set~$V(T(\gamma))$ is bounded by 
$1\cdot3\cdot\cdot5\cdot (2kM-2r-1)$. The above considerations 
provide the bound
$$
|\Gamma(k,M,r)|\le
\binom{2kM}{2r}1\cdot3\cdot5\cdots(2kM+2r-1)=\binom{2kM}{2r}
\frac{(2kM+2r)!}{2^{kM+r}(kM+r)!}. \tag13.5
$$
Stirling's formula yields that
$\frac{(2kM+2r)!}{2^{kM+r}(kM+r)!}\le A\(\frac2e\)^{kM+r}(kM+r)^{kM+r}$
with some constant $A>\sqrt2$ if $M\ge M_0$ with some $M_0=M_0(A)$.
Since $r\le kM$ we can write $(kM+r)^{kM+r}\le (kM)^{kM}\(1+\frac
r{kM}\)^{kM}(2kM)^r\le (kM)^{kM+r}e^r2^r$. The above calculation
together with~(13.5) imply inequality~(13.4).

\medskip
For a diagram $\gamma\in\Gamma(k,M,r)$ we have $W(\gamma)=2r$, 
and beside this the cardinality of the set~$U(\gamma)$ defined 
in the formulation of Lemma~11.3 satisfies the inequality 
$|U(\gamma)|\ge 2M-W(\gamma)=2M-2r$. Hence by relation~(11.18)
$n^{-W(\gamma)/2}|F_\gamma|\le 2^{2r} n^{-r}\sigma^{|U(\gamma)|}
\le 2^{2r} \(n\sigma^2\)^{-r}\sigma^{2M}\le\eta^{r}2^{2r}(kM)^{-r}
\sigma^{2M}$ for $\gamma\in\Gamma(k,M,r)$ if 
$kM\le \eta n\sigma^2$ and $\sigma^2\le1$.

This estimate together with relation~(11.17) imply that for 
$kM\le\eta n\sigma^2$
$$
E\(n^{-k/2}k!I_{n,k}(f_{k})\)^{2M}\le\sum_{\gamma\in\Gamma(k,M)}
n^{-W(\gamma)/2}\cdot |F_\gamma|\le\sum_{r=0}^{kM}|\Gamma(k,M,r)|
\eta^{r}2^{2r}(kM)^{-r}\sigma^{2M}.
$$

Hence by formula (13.4)
$$
\align
E\(n^{-k/2}k!I_{n,k}(f_{k})\)^{2M}&\le
A\(\frac2e\)^{kM}(kM)^{kM}\sigma^{2M}\sum_{r=0}^{kM}\binom{2kM}{2r}
\(4\sqrt{\eta}\)^{2r}\\
&\le A\(\frac2e\)^{kM}(kM)^{kM}\sigma^{2M}
\(1+4\sqrt{\eta}\)^{2kM}
\endalign
$$
if $kM_0\le kM\le\eta n\sigma^2$. Thus we have proved
Proposition~13.2 with $C=4$.

\medskip
It is not difficult to prove Theorem 8.5 with the help of
Proposition~13.1.

\medskip\noindent
{\it Proof of Theorem 8.5.}\/
By formula (13.2) which is a consequence of Proposition~13.1 and
the Markov inequality
$$
P\(|k!Z_{\mu,k}(f)|>u\)\le\frac{E\(k!Z_{\mu,k}(f)\)^{2M}}{u^{2M}}
\le A\(\frac {2kM\sigma^{2/k}}{eu^{2/k}}\)^{kM} \tag13.6
$$
with some constant $A>\sqrt2$ if $M\ge M_0$ with some constant
$M_0=M_0(A)$, and $M$ is an integer.

Put $\bar M=\bar M(u)=\frac1{2k}\(\frac u\sigma\)^{2/k}$, and
$M=M(u)=[\bar M]$, where $[x]$ denotes the integer part of a real
number $x$. Choose some number $u_0$ such that $\frac1{2k}\(\frac
{u_0}\sigma\)^{2/k}\ge M_0+1$. Then relation (13.6) can be applied
with $M=M(u)$ for $u\ge u_0$, and it yields that
$$
\aligned
P\(|k!Z_{\mu,k}(f)|>u\) &\le A\(\frac
{2kM\sigma^{2/k}}{eu^{2/k}}\)^{kM}\le e^{-kM}\le Ae^{k}e^{-k\bar M}\\
&=Ae^k\exp\left\{-\frac12
\(\frac u\sigma\)^{2/k}\right\} \quad\text{if } u\ge u_0.
\endaligned \tag13.7
$$
Relation (13.7) means that relation (8.14) holds for $u\ge u_0$ with
the pre-exponential coefficient $Ae^k$. By enlarging this
coefficient if it is needed it can be guaranteed that
relation~(8.14) holds for all $u>0$. Theorem~8.5 is proved.

\medskip
Theorem 8.3 can be proved similarly by means of Proposition~13.2.
Nevertheless, the proof is technically more complicated, since
in this case the optimal choice of the parameter in the Markov
inequality cannot be given in such a direct form as in the proof of
Theorem~8.5. In this case the Markov inequality is applied with an
only almost optimal choice of the parameter~$M$.

\medskip\noindent
{\it Proof of Theorem 8.3.}\/ The Markov inequality and
relation~(13.3) with $\eta=\frac{kM}{n\sigma^2}$ imply that
$$
\aligned
P(k!n^{-k/2}|I_{n,k}(f)|>u)&\le
\frac{E\(k!n^{-k/2}I_{n,k}(f)\)^{2M}}{u^{2M}}\\
&\le A\(\frac1e\cdot 2kM\(1+C\frac{\sqrt{kM}}{\sqrt n\sigma}\)^2
\(\frac\sigma u\)^{2/k}\)^{kM}
\endaligned \tag13.8
$$
for all integers $M\ge M_0$ with some $M_0=M_0(A)$.

Relation (8.10) will be proved with the help of estimate (13.8)
first in the case $D\le\frac u\sigma \le n^{k/2}\sigma^k$ with a
sufficiently large constant $D=D(k,C)>0$ depending on $k$ and the
constant $C$ in (13.8). To this end let us introduce the number
$\bar M$ by means of the formula
$$
k\bar M=\frac12\(\frac u\sigma\)^{2/k}\frac1
{1+B\frac{ \(\frac u\sigma\)^{1/k}}{\sqrt n\sigma}}
=\frac12\(\frac u\sigma\)^{2/k}\frac1
{1+B\(u n^{-k/2}\sigma^{-(k+1)}\)^{1/k}}
$$
with a sufficiently large number $B=B(C)>0$ and $M=[\bar M]$,
where $[x]$ means the integer part of the number $x$.

Observe that $\sqrt{k\bar M}\le\(\frac u\sigma\)^{1/k}$, $\frac{\sqrt
{k\bar M}}{\sqrt n\sigma}\le\(un^{-k/2}\sigma^{-(k+1)}\)^{1/k}\le1$,
and
$$
\(1+C\frac{\sqrt{k\bar M}}{\sqrt n\sigma}\)^2\le
1+B\frac{\sqrt{k\bar M}}{\sqrt n\sigma}\le 1+B\(u n^{-k/2}
\sigma^{-(k+1)}\)^{1/k}
$$
with a sufficiently large $B=B(C)>0$ if
$\frac u\sigma\le n^{k/2}\sigma^k$.  Hence
$$
\aligned
\frac1e\cdot 2kM\(1+C\frac{\sqrt{kM}}{\sqrt n\sigma}\)^2
\(\frac\sigma u\)^{2/k}&\le
\frac1e\cdot 2k\bar M\(1+C\frac{\sqrt{k\bar M}}{\sqrt n\sigma}\)^2
\(\frac\sigma u\)^{2/k}  \\
&=\frac1e\cdot\frac{\(1+C\frac{\sqrt{k\bar M}}{\sqrt n\sigma}\)^2}
{1+B\(u n^{-k/2}\sigma^{-(k+1)}\)^{1/k}}\le\frac1e
\endaligned \tag13.9
$$
if $\frac u\sigma\le n^{k/2}\sigma^k$. If the inequality
$D\le\frac u\sigma$ also holds with a sufficiently large
$D=D(B,k)>0$, then $M\ge M_0$, and the conditions of
inequality~(13.8) hold. This inequality together with
inequality~(13.9) yield that
$$
P(k!n^{-k/2}|I_{n,k}(f)|>u)\le A e^{-kM}\le Ae^k e^{-k\bar M}
$$
if
$D\le\frac u\sigma\le n^{k/2}\sigma^k$, i.e.\ inequality (8.10)
holds in this case with a pre-exponential constant $Ae^k$. In the 
case $\frac u\sigma\le D$ the right-hand side of~(8.10) is larger 
than~1 if we choose the pre-exponential term~$A$ sufficiently 
large. Hence inequality~(8.10) holds for all 
$0\le\frac u\sigma\le n^{k/2}\sigma^k$ with a sufficiently large
pre-exponential term~$A$. Theorem~8.3 is proved.

\medskip
Example 8.7 is a relatively simple consequence of It\^o's formula
for multiple Wiener--It\^o integrals.

\medskip\noindent
{\it Proof of Example 8.7.}\/ We may restrict our attention to the
case $k\ge2$. It\^o's formula for multiple Wiener-It\^o integrals,
more explicitly relation~(10.21), implies that the random
variable $k!Z_{\mu,k}(f)$ can be expressed as $k!Z_{\mu,k}(f)
=\sigma H_k\(\int f_0(x)\mu_W(\,dx)\)=\sigma H_k(\eta)$, where
$H_k(x)$ is the $k$-th Hermite polynomial with leading
coefficient~1, and $\eta=\int f_0(x)\mu_W(\,dx)$ is a standard
normal random variable. Hence we get by exploiting that the
coefficient of $x^{k-1}$ in the polynomial $H_k(x)$ is zero that
$P(k!|Z_{\mu,k}(f)|>u)=P(|H_k(\eta)|\ge\frac u\sigma)\ge
P\(|\eta^k|-D|\eta^{k-2}|>\frac u\sigma\)$ with a sufficiently
large constant $D>0$ if $\frac u\sigma>1$. There exist such
positive constants $A$ and $B$ that
$$
P\(|\eta^k|-D|\eta^{k-2}|>\frac u\sigma\)
\ge P\(|\eta^k|>\frac u\sigma+A\(\frac u\sigma\)^{(k-2)/k}\)\quad
\text{if } \frac u\sigma>B.
$$
Hence
$$
P(k!|Z_{\mu,k}(f)|>u)\ge
P\(|\eta|>\(\frac u\sigma\)^{1/k}\(1+A\(\frac u\sigma\)^{-2/k}\)\)
\ge\frac{\bar C \exp\left\{-\frac12\(\frac u\sigma\)^{2/k}\right\}}
{\(\frac u\sigma\)^{1/k}+1}
$$
with an appropriate $\bar C>0$ if $\frac u\sigma>B$. Since
$P(k!|Z_{\mu,k}(f)|>0)>0$, the above inequality also holds
for $0\le \frac u\sigma\le B$ if the constant $\bar C>0$ is chosen
sufficiently small. This means that relation (8.16) holds.

\medskip
Next we prove a multivariate version of Hoeffding's inequality.
Before its formulation some notations will be introduced.

Let us fix two positive integers~$k$ and~$n$ and some
real numbers $a(j_1,\dots,j_k)$ for all sequences of arguments
$\{j_1,\dots,j_k\}$ such that $1\le j_l\le n$, $1\le l\le k$, and
$j_l\neq j_{l'}$ if $l\neq l'$.

With the help of the above real numbers $a(\cdot)$ and a
sequence of independent random variables $\e_1,\dots,\e_n$,
$P(\e_j=1)=P(\e_j=-1)=\frac12$, $1\le j\le n$, the random
variable
$$
V=\sum\Sb (j_1,\dots, j_k)\colon\; 1\le j_l\le n \text{ for all }
1\le l\le k,\\
j_l\neq j_{l'} \text{ if }l\neq l' \endSb a(j_1,\dots, j_k)
\e_{j_1}\cdots \e_{j_k} \tag13.10
$$
and number
$$
S^2=\sum\Sb (j_1,\dots, j_k)\colon\; 1\le j_l\le n \text{ for all }
1\le l\le k,\\
j_l\neq j_{l'} \text{ if }l\neq l' \endSb
a^2(j_1,\dots, j_k). \tag13.11
$$
will be introduced.

With the help of the above notations the following result can be
formulated.

\medskip\noindent
{\bf Theorem 13.3. (The multivariate version of Hoeffding's
inequality).} {\it The random variable $V$ defined in
formula~(13.10) satisfies the inequality
$$
P(|V|>u)\le C \exp\left\{-\frac12\(\frac uS\)^{2/k}\right\}
\quad\text{for all }u\ge 0 \tag13.12
$$
with the constant $S$ defined in (13.11) and some constants $C>0$
depending only on the parameter $k$ in the expression $V$.}

\medskip
Theorem~13.3 will be proved by means of two simple lemmas. Before
their formulation the random variable
$$
Z=\sum\Sb (j_1,\dots, j_k)\colon\; 1\le j_l\le n \text{ for all } 1\le
l\le k,\\
j_l\neq j_{l'} \text{ if }l\neq l'\endSb |a(j_1,\dots,j_k)|
\eta_{j_1}\cdots \eta_{j_k} \tag13.13
$$
will be introduced, where $\eta_1,\dots,\eta_n$ are independent
random variables with standard normal distribution, and the numbers
$a(j_1,\dots,j_k)$ agree with those in formula (13.10). The
following lemmas will be proved.

\medskip\noindent
{\bf Lemma 13.4.} {\it The random variables $V$ and $Z$ introduced
in (13.10) and (13.13) satisfy the inequality
$$
EV^{2M}\le EZ^{2M}\quad\text{for all }M=1,2,\dots. \tag13.14
$$
}

\medskip\noindent
{\bf Lemma 13.5.} {\it The random variable $Z$ defined in formula
(13.13) satisfies the inequality
$$
EZ^{2M}\le 1\cdot3\cdot5\cdots(2kM-1)S^{2M}\quad\text{for all }
M=1,2,\dots \tag13.15
$$
with the constant $S$ defined in formula (13.11).}

\medskip\noindent
{\it Proof of Lemma 13.4.}\/ We can write, by carrying out the
multiplications in the expressions $EV^{2M}$ and $EZ^{2M}$,
by exploiting the additive and multiplicative properties of the
expectation for sums and products of independent random variables
together with the identities $E\e_j^{2k+1}=0$ and $E\eta_j^{2k+1}=0$
for all $k=0,1,\dots$  that
$$
EV^{2M}=  \!\!\!\!\!\!\!\!\!
\sum\Sb (j_1,\dots, j_l,\, m_1,\dots, m_l)\colon \\
1\le j_s\le n,\;
m_s\ge1,\; 1\le s\le l,\; m_1+\dots+m_l=kM\endSb \!\!\!\!\!\!\!\!\!
A(j_1,\dots,j_l,m_1,\dots,m_l)E\e_{j_1}^{2m_1}\cdots E\e_{j_l}^{2m_l}
\tag13.16
$$
and
$$
EZ^{2M}= \!\!\!\!\!\!\!\!\!
\sum \Sb (j_1,\dots, j_l,\, m_1,\dots, m_l)\colon \\
1\le j_s\le n,\;
m_s\ge1,\; 1\le s\le l,\; m_1+\dots+m_l=kM\endSb \!\!\!\!\!\!\!\!\!
B(j_1,\dots,j_l,m_1,\dots,m_l) E\eta_{j_1}^{2m_1}\cdots
E\eta_{j_l}^{2m_l} \tag13.17
$$
with some coefficients $A(j_1,\dots,j_l,m_1,\dots,m_l)$ and
$B(j_1,\dots,j_l,m_1,\dots,m_l)$ such that
$$|
A(j_1,\dots,j_l,m_1,\dots,m_l)|\le
B(j_1,\dots,j_l,m_1,\dots,m_l). \tag13.18
$$
The coefficients $A(\cdot,\cdot,\cdot)$ and $B(\cdot,\cdot,\cdot)$
could be expressed explicitly, but we do not need such a formula.
What is important for us is that $A(\cdot,\cdot,\cdot)$ can be
expressed as the sum of certain terms, and $B(\cdot,\cdot,\cdot)$
as the sum of the absolute value of the same terms. Hence
relation~(13.18) holds. Since $E\e_j^{2m}\le E\eta_j^{2m}$ for all
parameters $j$ and $m$ formulas (13.16), (13.17) and (13.18) imply
Lemma~13.4.

\medskip\noindent
{\it Proof of Lemma~13.5.} Let us consider a white noise $W(\cdot)$
on the unit interval $[0,1]$ with the Lebesgue measure $\lambda$ on
$[0,1]$ as its reference measure, i.e.\ let us take a set of
Gaussian random variables $W(A)$ indexed by the measurable sets
$A\subset [0,1]$ such that $EW(A)=0$, $EW(A)W(B)=\lambda(A\cap B)$
with the Lebesgue measure $\lambda$ for all measurable subsets of
the interval $[0,1]$. Let us introduce $n$ orthonormal functions
$\varphi_1(x),\dots,\varphi_n(x)$ with respect to the Lebesgue
measure on the interval $[0,1]$, and define the random variables
$\eta_j=\int \varphi_j(x)W(\,dx)$, $0\le j\le n$. Then
$\eta_1,\dots,\eta_n$ are independent random variables with standard
normal distribution, hence we may assume that they appear in the
definition of the random variable~$Z$ in formula (13.13). Beside
this, the identity $\eta_{j_1}\cdots\eta_{j_k}=\int \varphi_{j_1}(x_1)
\cdots\varphi_{j_k}(x_k)W(\,dx_1)\dots W(\,dx_k)$ holds for all
$k$-tuples $(j_1,\dots,j_k)$, such that $1\le j_s\le n$ for all
$1\le s\le k$, and the indices $j_1$,\dots, $j_s$ are different.
This identity follows from It\^o's formula for multiple Wiener--It\^o
integrals formulated in formula~(10.20) of Theorem~10.3.

Hence the random variable $Z$ defined in~(13.13) can be written in
the form
$$
Z=\int f(x_1,\dots,x_k)W(\,dx_1)\dots W(\,dx_k)
$$
with the function
$$
f(x_1,\dots,x_k)=
\sum\Sb (j_1,\dots, j_k)\colon\; 1\le j_l\le n \text{ for all } 1\le
l\le k,\\
j_l\neq j_{l'} \text{ if }l\neq l'\endSb |a(j_1,\dots,j_k)|
\varphi_{j_1}(x_1)\cdots \varphi_{j_k}(x_k).
$$
Because of the orthogonality of the functions $\varphi_j(x)$
$$
S^2=\int_{[0,1]^k} f^2(x_1,\dots,x_k)\,dx_1\dots\,dx_k.
$$
Lemma~13.5 is a straightforward consequence of the above relations
and formula~(13.1) in Proposition~13.1.

\medskip\noindent
{\it Proof of Theorem~13.3.}\/ The proof of Theorem~13.3 with the
help of Lemmas~13.4 and~13.5 is an almost word for word repetition
of the proof of Theorem~8.5. By Lemma~13.4 inequality~(13.15)
remains valid if the random variable $Z$ is replaced by the random
variable~$V$ at its left-hand side. Hence the Stirling formula
yields that
$$
EV^{2M}\le EZ^{2M}\le \frac{(2kM)!}{2^{kM}(kM)!} S^{2M}\le C
\(\frac2e\)^{kM}(kM)^{kM}S^{2M}
$$
for any $C\ge\sqrt2$ if $M\ge M_0(A)$. As a consequence, by the
Markov inequality the estimate
$$
P(|V|>u)\le\frac{EV^{2M}}{u^{2M}}\le C\(\frac{2kM}e\(\frac
Su\)^{2/k}\)^{kM}     \tag13.19
$$
holds for all $C>\sqrt 2$ if $M\ge M_0(C)$. Put $k\bar M=k\bar
M(u)=\frac12\(\frac uS\)^{2/k}$ and $M=M(u)=[\bar M]$, where
$[x]$ denotes the integer part of the number~$x$. Let us choose
a threshold number $u_0$ by the identity
$\frac1{2k}\(\frac{u_0}S\)^{2/k}=M_0(C)+1$.
Formula~(13.19) can be applied with $M=M(u)$ for $u\ge u_0$, and
it yields that
$$
P(|V|>u)\le Ce^{-kM}\le Ce^ke^{-k\bar M}=Ce^k\exp\left\{-\frac12
\(\frac uS\)^{2/k}\right\}\ \quad\text{if } u\ge u_0.
$$
The last inequality means that relation (13.12) holds for $u\ge u_0$
if the constant $C$ is replaced by $Ce^k$ in it. With the choice of
a sufficiently large constant~$C$ relation (13.12) holds
for all $u\ge0$. Theorem~13.3 is proved.

\medskip\noindent
{\script  13. B) A short discussion about the methods and results.}

\medskip\noindent
A comparison of Theorem 8.5 and Example 8.7 shows that the estimate
(8.15) is sharp. At least no essential improvement of this estimate
is possible which holds for {\it all}\/ Wiener--It\^o integrals 
with a kernel function $f$ satisfying the conditions of Theorem~8.5. 
This fact also indicates that the bounds~(13.1) and~(13.2) on high 
moments of Wiener--It\^o integrals are sharp. It is worth 
while comparing formula (13.2) with the estimate of 
Proposition~13.2 on moments of degenerate $U$-statistics.

Let us consider a normalized $k$-fold degenerate $U$-statistic
$n^{-k/2}k!I_{n,k}(f)$ with some kernel function $f$ and a
$\mu$-distributed sample of size~$n$. Let us compare its moments 
with those of a $k$-fold Wiener--It\^o integral k!$Z_{\mu,k}(f)$
with the same kernel function~$f$ with respect to a white noise
$\mu_W$ with reference measure~$\mu$. Let $\sigma$ denote the
$L_2$-norm of the kernel function~$f$. If $M\le\e n\sigma^2$ with 
a small number $\e>0$, then Proposition~13.2 (with an appropriate 
choice of the parameter~$\eta$ which is small in this case) 
provides an almost as good bound on the $2M$-th moment of the
normalized $U$-statistic as Proposition~13.1 provides on the 
$2M$-th moment of the corresponding Wiener--It\^o integral. In 
the case $M\le Cn\sigma^2$ with some fixed (not necessarily small) 
number $C>0$ the $2M$-th moment of the normalized $U$-statistic 
can be bounded by $C(k)^M$ times the natural estimate on the 
$2M$-th moment of the Wiener--It\^o integral with some 
constant~$C(k)>0$ depending only on the number~$C$. This can be
so interpreted that in this case the estimate on the moments of the
normalized $U$-statistic is weaker than the estimate on the moments
of the Wiener--It\^o integral, but they are still comparable.
Finally, in the case $M\gg n\sigma^2$ the estimate on the $2M$-th
moment of the normalized $U$-statistic is much worse than the
estimate on the $2M$-th moment of the Wiener--It\^o integral.

A similar picture arises if the distribution of the normalized 
degenerate $U$-statistic
$$
F_n(u)=P(n^{-k/2}k!|I_{n,k}(f)|>u)
$$
is compared to the distribution of the Wiener--It\^o integral
$$
G(u)=P(k!|Z_{\mu,k}(f)|>u).
$$
A comparison of Theorems~8.3 and~8.5 shows that for
$0\le u\le\e n^{k/2}\sigma^{k+1}$ with a small $\e>0$
an almost as good estimate holds $F_n(u)$ as for $G(u)$. In the 
case $0\le u\le n^{k/2}\sigma^{k+1}$ the behaviour of $F_n(u)$ 
and $G(u)$ is similar, only in the exponent of the estimate on 
$F_n(u)$ in formula (8.10) a worse constant appears. Finally, if 
$u\gg n^{k/2}\sigma^{k+1}$, then --- as Example~8.8 shows, at 
least in the case $k=2$, --- the  (tail) distribution function 
$F_n(u)$ satisfies a much worse estimate than the function 
$G(u)$. Thus a similar picture arises as in the case when the 
estimate on the tail-distribution of normalized sums of independent 
random variables, discussed in Section~3, is compared to the 
behaviour of the standard normal distribution in the neighbourhood 
of infinity. To understand this similarity better it is useful to
recall Theorem~10.4, the limit theorem about normalized degenerate
$U$-statistics. Theorems 8.3 and~8.5 enable us to compare the tail
behaviour of normalized degenerate $U$-statistics with their limit
presented in the form of multiple Wiener--It\^o integrals, while
the one-variate versions of these results compare the distribution
of sums of independent random variables with their Gaussian limit.

The above results show that good bounds on the moments of degenerate
$U$-statistics and multiple Wiener--It\^o also provide a good
estimate on their distribution. To understand the behaviour of high
moments of degenerate $U$-statistics it is useful to have a closer 
look at the simplest case $k=1$, when the moments of sums of 
independent random variables with expectation zero are considered.

Let us consider a sequence of independent and identically
distributed random variables $\xi_1,\dots,\xi_n$ with expectation
zero, take their sum $S_n=\sum\limits_{j=1}^n\xi_j$, and let us try
to give a good estimate on the moments $ES_n^{2M}$ for all
$M=1,2,\dots$. Because of the independence of the random variables
$\xi_j$ and the condition $E\xi_j=0$ the identity
$$
ES_n^{2M}=\sum\Sb (j_1,\dots,j_s,l_1,\dots,l_s)\\
j_1+\dots+j_s=2M,\,j_u\ge 2,\text{ for all }1\le u\le s \\
l_u\neq l_{u'} \text { if }u\neq u'\endSb
E\xi_{l_1}^{j_1}\cdots E\xi_{l_s}^{j_s}
\tag13.20
$$
holds. Simple combinatorial considerations show that a dominating
number of terms at the right-hand side of (13.20) are indexed by a
vector $(j_1,\dots,j_M;\,l_1,\dots,l_M)$ such that $j_u=2$ for all
$1\le u\le M$, and the number of such vectors is equal to $\binom nM
\frac{(2M)!}{2^M}\sim n^M\frac{(2M)!}{2^MM!}$. The last asymptotic
relation holds if the number $n$ of terms in the random sum~$S_n$
is sufficiently large. The above considerations suggest that under
not too restrictive conditions $ES_n^{2M}\sim
\left(n\sigma^2\right)^M\frac{(2M)!}{2^MM!}=E\eta_{n\sigma^2}^{2M}$,
where $\sigma^2=E\xi^2$ is the variance of the terms in the sum
$S_n$, and $\eta_u$ denotes a random variable with normal distribution
with expectation zero and variance~$u$. The question arises when
the above heuristic argument gives a right estimate.

For the sake of simplicity let us restrict our attention to the
case when the absolute value of the random variables $\xi_j$ is
bounded by~1. Let us observe that even in this case the above
heuristic argument holds only under the condition that the variance
$\sigma^2$ of the random variables $\xi_j$ is not too small.
Indeed, let us consider such random variables $\xi_j$, for which
$P(\xi_j=1)=P(\xi_j=-1)=\frac{\sigma^2}2$, $P(\xi_j=0)=1-\sigma^2$.
Then these random variables $\xi_j$ have variance $\sigma^2$, and
the contribution of the terms $E\xi_j^{2M}$, $1\le j\le n$, to the
sum in (13.20) equals $n\sigma^2$. If $\sigma^2$ is very small,
then it may happen that $n\sigma^2\gg\left(n\sigma^2\right)^M
\frac{(2M)!}{2^MM!}$, and the approximation given for $ES_n^{2M}$
in the previous paragraph does not hold any longer. Hence the
asymptotic relation for a very high moment $ES_n^{2M}$ suggested
by the above heuristic argument may only hold if the variance
$\sigma^2$ of the summands satisfies an appropriate lower bound.

In the proof of Proposition~13.2 a similar picture appears in a
hidden way. In the calculation of the moments of a degenerate 
$U$-statistic the contribution of certain (closed) diagrams, 
more precisely of some integrals defined with their help, has to 
be estimated. Some of these diagrams (those in which all chains
have length~2) appear also in the calculation of the moments of 
multiple Wiener--It\^o integrals. In the calculation of the 
moments of sums of independent random variables the terms 
consisting of products of second moments play such a role in 
the sum in formula~(13.20) as the `nice' diagrams consisting of 
chains of length~2 play in the calculation of the moments of 
degenerate $U$-statistics in formula~(11.17). In nice cases the 
remaining diagrams do not give a much greater contribution than 
these `nice' diagrams, and we get an almost as good bound for 
the moments of a normalized degenerate $U$-statistic as for the 
moments of the corresponding multiple Wiener--It\^o integral. 
The proof of Proposition~13.2 shows that such a situation 
appears under very general conditions.

Let me also remark that there is an essential difference 
between the tail behaviour of Wiener--It\^o integrals and 
normalized degenerate $U$-statistics. A good estimate can be
given on the tail distribution of Wiener--It\^o integrals which
depends only on the $L_2$-norm of the kernel function, while in
the case of normalized degenerate $U$-statistics the 
corresponding estimate depends not only on the $L_2$-norm but 
also on the $L_\infty$ norm of the kernel function. In 
Theorem~8.3 such an estimate is proved. 

For $k\ge2$ the distribution of $k$-fold Wiener-It\^o integrals are
not determined by the $L_2$-norm of their kernel functions. This is
an essential difference between Wiener--It\^o integrals of order
$k\ge2$ and $k=1$. In the case $k=1$ a Wiener--It\^o integral is
a Gaussian random variable with expectation zero, and its variance
equals the square of the $L_2$-norm of its kernel function. Hence
its distribution is completely determined by the $L_2$-norm of its
kernel function. On the other hand, the distribution of a
Wiener--It\^o integral of order $k\ge2$ is not determined by its
variance. Theorem~8.5 yields a `worst case' estimate on the
distribution of Wiener--It\^o integrals if we have a bound on their
variance. In the statistical problems which provided the main
motivation for this work such estimates are needed, but it may be
interesting to know what kind of estimates are known about the
distribution of a multiple Wiener--It\^o integral or degenerate
$U$-statistic if we have some additional information about its
kernel function. Some results will be mentioned in this direction,
but most technical details will be omitted from their discussion.

H. P. Mc. Kean proved the following lower bound on the distribution
of multiple Wiener--It\^o integrals. (See [29] or [42].)

\medskip\noindent
{\bf Theorem 13.6. (Lower bound on the distribution of
Wiener--It\^o integrals).} {\it All $k$-fold Wiener--It\^o integrals
$Z_{\mu,k}(f)$ satisfy the inequality
$$
P(|Z_{\mu,k}(f)|>u)>Ke^{-Au^{2/k}} \tag13.21
$$
with some numbers $K=K(f,\mu)>0$ and $A=A(f,\mu)>0$.}

\medskip\noindent
The constant $A$ in the exponent $Au^{2/k}$ of formula~(13.21) is 
always finite, but Mc.~Kean's proof yields no explicit upper 
bound on it. The following example shows that in certain cases 
if we fix the constant~$K$ in relation~(13.21), then this 
inequality holds only with a very large constant $A>0$ even 
if the variance of the Wiener--It\^o integral equals~1.

Take a probability measure $\mu$ and a white noise $\mu_W$ with
reference measure $\mu$ on a measurable space $(X,\Cal X)$, and let
$\varphi_1,\varphi_2,\dots$ be a sequence of orthonormal functions
on $(X,\Cal X)$ with respect to this measure $\mu$. Define for all
$L=1,2,\dots$, the function
$$
f(x_1,\dots,x_k)=f_L(x_1,\dots,x_k)=(k!)^{1/2}L^{-1/2}\summ_{j=1}^L
\varphi_j(x_1)\cdots\varphi_j(x_k) \tag13.22
$$
and the Wiener--It\^o integral
$$
Z_{\mu,k}(f)=Z_{\mu,k}(f_L)=\frac1{k!}\int f_L(x_1,\dots,x_k)
\mu_W(\,dx_1)\dots\mu_W(\,dx_k).
$$
Then $EZ_{\mu,k}^2(f)=1$, and the high moments of $Z_{\mu,k}(f)$ can
be well estimated. For a large parameter~$L$ these moments are much
smaller, than the quantities suggested by Proposition~13.1. (The
calculation leading to the estimation of the moments of
$Z_{\mu,k}(f)$ will be omitted.) These moment estimates also imply 
that if the parameter~$L$ is large, then for not too large 
numbers~$u$ the probability $P(|Z_{\mu,k}(f)|>u)$ has a much better
estimate than that given in Theorem~8.5. As a consequence, 
for a large number $L$ and fixed number~$K$ relation~(13.21) may 
hold only with a very big number $A>0$.

We can expect that if we take a Gaussian random 
polynomial~$P(\xi_1,\dots,\xi_n)$ whose arguments are Gaussian
random variables $\xi_1,\dots,\xi_n$, and which is the sum of
many small almost independent terms, then 
a similar picture arises as in the case of a Wiener--It\^o 
integral with kernel function~(13.22) with a large parameter~$L$. 
Such a random polynomial has an almost Gaussian distribution by 
the central limit theorem, and we can also expect that its not 
too high moments behave so as the corresponding moments of a 
Gaussian random variable with expectation zero and the same 
variance as the Gaussian random polynomial we consider. Such a
bound on the moments has the consequence that the estimate on
the probability $(P(\xi_1,\dots,\xi_n)>u)$ given in Theorem~8.5
can be improved if the number~$u$ is not too large. A similar 
picture arises if we consider Wiener--It\^o integrals whose 
kernel function satisfies some `almost independence' properties. 
The problem is to find the right properties under which we can 
get a good estimate that exploits the almost independence
property of a Gaussian random polynomial or of a Wiener--It\^o 
integral. The main result of R.~Lata{\l}a's paper~[26] can be 
considered as a response to this question. I describe this 
result below. 

To formulate Lata{\l}a's result some new notions have to be 
introduced.  Given a finite set $A$ let ${\Cal P}(A)$ denote the 
set of all its partitions. If a partition
$P=\{B_1,\dots,B_s\}\in{\Cal P}(A)$ consists of $s$ elements then we
say that this partition has order~$s$, and write $|P|=s$. In the
special case $A=\{1,\dots,k\}$ the notation ${\Cal P}(A)={\Cal P}_k$
will be used. Given a measurable space $(X,{\Cal X})$ with a
probability measure $\mu$ on it together with a finite set
$B=\{b_1,\dots,b_j\}$ let us introduce the following notations. Take
$j$ different copies $(X_{b_r},{\Cal X}_{b_r})$ and $\mu_{b_r}$,
$1\le r\le j$, of this measurable space and probability measure indexed
by the elements of the set $B$, and define their product
$(X^{(B)},{\Cal X}^{(B)},\mu^{(B)})=\left(\prod\limits_{r=1}^j X_{b_r},
\prod\limits_{r=1}^j{\Cal X}_{b_r},
\prod\limits_{r=1}^j\mu_{b_r}\right)$. The points
$(x_{b_1},\dots,x_{b_j})\in X^{(B)}$ will be denoted by
$x^{(B)}\in X^{(B)}$ in the sequel. With the help  of the above
notations I introduce the quantities needed in the formulation of the
following Theorem~13.7.

Let a function $f=f(x_1,\dots,x_k)$ be given on the $k$-fold product
$(X^k,{\Cal X}^k,\mu^k)$ of a measurable space $(X,{\Cal X})$ with a
probability measure $\mu$. For all partitions
$P=\{B_1,\dots,B_s\}\in{\Cal P}_k$ of the set $\{1,\dots,k\}$ consider
the functions $g_r\left(x^{(B_r)}\right)$ on the space $X^{(B_r)}$,
$1\le r\le s$, and define with their help the quantities
$$
\aligned
 \alpha(P)&=\alpha(P,f,\mu) 
 =\sup_{g_1,\dots,g_s} \int f(x_1,\dots,x_k)
g_1\left(x^{(B_1)}\right)\cdots g_s\left(x^{(B_s)}\right)\mu(dx_1)
\dots\mu(dx_k); \\
&\qquad \text{where supremum is taken for such functions } 
g_1,\dots,g_s,\quad g_r\colon\; X^{B_r}\to R^1 \\
&\qquad\qquad \text{for which } 
\int g_r^2\left(x^{(B_r)}\right)\mu^{(B_r)}\left(\,dx^{(B_r)}\right)\le1
\quad \text{for all } 1\le r\le s, 
\endaligned \tag13.23
$$
and put
$$
\alpha_s=\max_{P\in\Cal P_k,\,|P|=s}\alpha(P), \quad 1\le s\le k. \tag13.24
$$
In Lata{\l}a's estimation of Wiener--It\^o integrals of order~$k$ 
the quantities $\alpha_s$, $1\le s\le k$, play a similar role as 
the number $\sigma^2$ in Theorem~8.5. Observe that in the case 
$|P|=1$, i.e.\ if $P=\{1,\dots,k\}$ the identity 
$\alpha^2(P)=\int f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)$ 
holds, which means that $\alpha_1=\sigma$. The following estimate 
is valid for Wiener--It\^o integrals of general order.

\medskip\noindent
{\bf Theorem 13.7. (Lata{\l}a's estimate about the tail-distribution
of Wiener--It\^o integrals).} {\it Let a $k$-fold Wiener--It\^o integral
$Z_{\mu,k}(f)$, $k\ge1$, be defined with the help of a white noise
$\mu_W$ with a non-atomic reference measure~$\mu$ and a kernel
function $f$ of $k$-variable such that
$\int f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)<\infty$. There is
some universal constant $C(k)<\infty$ depending only of the order~$k$
of the random integral such that the inequalities
$$
E(Z_{\mu,k}(f))^{2M}\le \(C(k)\max_{1\le s\le k}(M^{s/2}\alpha_s)\)^{2M},
\tag13.25
$$
and
$$
P(|Z_{\mu,k}(f)|>u)\le C(k)\exp\left\{-\frac1{C(k)}\min_{1\le s\le k}
\left(\frac u{\alpha_s}\right)^{2/s}\right\} \tag13.26
$$
hold for all $M=1,2,\dots$ and $u>0$ with the quantities $\alpha_s$, 
defined in formulas~(13.23) and~(13.24).}

\medskip
Inequality~(13.26) is a simple consequence of~(13.25). In the special 
case when $\alpha_s\le M^{-(s-1)/2}$ for all $1\le s\le k$, then 
inequality~(13.25) says that the moment $EZ_{\mu,k}(f)^{2M}$ has 
the same magnitude as the $2M$-th moment of a standard Gaussian 
random variable multiplied by a constant, and it implies a good
estimate on $P(|Z_{\mu,k}(f)|>u)$ given in~(13.26). Actually the
result of Theorem~13.7 can be reduced to the special case when
$\alpha_s\le M^{-(s-1)/2}$ for all $1\le s\le k$. Thus it can be 
interpreted so that if the quantities~$\alpha_s$ of a $k$-fold 
Wiener--It\^o integral are sufficiently small, then these `almost 
independence' conditions imply that the $2M$-th moment of this 
integrals behaves like a one-fold Wiener--It\^o integral with the 
same variance.

Actually Lata{\l}a formulated his result in a different form, and
he proved a slightly weaker result. He considered Gaussian
polynomials of the following form: 
$$
P(\xi_j^{(s)},\;1\le j\le n,\,1\le s\le k)=\frac1{k!}
\sum_{ (j_1,\dots,j_k)\colon\,1\le j_s\le n,\,1\le s\le k}
a(j_1,\dots,j_k)\xi^{(1)}_{j_1}\cdots\xi^{(k)}_{j_k}, \tag13.27
$$
where $\xi_j^{(s)}$, $1\le j\le n$ and $1\le s\le k$, are independent
standard normal random variables. Lata{\l}a gave an estimate about
 the moments and tail-distribution of such random polynomials.

The problem about the behaviour of such random polynomials can be
reformulated as a problem about the behaviour of Wiener--It\^o 
integrals in the following way: Take a measurable space $(X,\Cal X)$
with a non-atomic measure~$\mu$ on it. Let $Z_\mu$ be a white noise
with reference measure~$\mu$, let us choose a set of orthogonal
functions $h^{(s)}_j(x)$, $1\le j\le n$, $1\le s\le k$, on the
space $(X,\Cal X)$ with respect to the measure~$\mu$, and define
the function
$$
f(x_1,\dots,x_k)=\frac1{k!}
\sum_{ (j_1,\dots,j_k)\colon\,1\le j_s\le n,\,1\le s\le k}
a(j_1,\dots,j_k)h^{(1)}_{j_1}(x_1)\cdots h^{(k)}_{j_k}(x_k) 
\tag13.28
$$
together with the Wiener--It\^o integral $Z_{\mu,k}(f)$. Since
the random integrals $\bar\xi_j^{(s)}=\int h_j^{(s)}(x)Z_\mu(\,dx)$,
$1\le j\le n$, $1\le s\le k$, are independent, standard Gaussian
random variables, it is not difficult to see with the help of 
It\^o's formula (Theorem~10.3 in this work) that the distributions 
of the random  polynomial
$P(\xi_j^{(s)},\;1\le j\le n,\,1\le s\le k)$ and $Z_{\mu,k}(f)$
agree. Here we reformulated Lata{\l}a's estimates about random
polynomials of the form~(13.27) to estimates about Wiener--It\^o
integrals with kernel function of the form~(13.28).

These estimates are equivalent to Lata{\l}a's result if we restrict
our attention to the special class of Wiener--It\^o integrals with 
kernel functions of the form~(13.28). But we have formulated our
result for Wiener--It\^o integrals with a general kernel function.
Lata{\l}a's proof heavily exploits the special structure of the
random polynomials given in~(13.27), the independence of the
random variables~$\xi_j^{(s)}$ for different parameters~$s$ in 
it. (It would be interesting to find a proof which does not 
exploit this property.) On the other hand, this result can 
be generalized to the case discussed in Theorem~13.7. This 
generalization can be proved by exploiting the theorem of 
de la Pe{\~n}a and Montgomery--Smith about the comparison of 
$U$-statistics and decoupled $U$-statistics (formulated in 
Theorem~14.3 of this work) and the properties of the 
Wiener--It\^o integrals. I omit the details of the proof.

Lata{\l}a also proved a converse estimate in~[26] about random
polynomials of Gaussian random polynomials which shows that the 
estimates of Theorem~13.7 are sharp. We formulate it in its
original form, i.e. we restrict our attention to the case of
Wiener--It\^o integrals with kernel functions of the form~(13.28). 

\medskip\noindent
{\bf Theorem 13.8. (A lower bound about the tail distribution of
Wiener--It\^o integrals).} {\it A random integral $Z_{\mu,k}(f)$
with a kernel function of the form~(13.28) satisfies the 
inequalities
$$
E(Z_{\mu,k}(f))^{2M}\ge 
\(C(k)\max_{1\le s\le k}(M^{s/2}\alpha_s)\)^{2M},
$$
and
$$
P(|Z_{\mu,k}(f)|>u)\ge \frac1{C(k)}\exp\left\{-C(k)
\min_{1\le s\le k}\left(\frac u{\alpha_s} \right)^{2/s}\right\}
$$
for all $M=1,2,\dots$ and $u>0$ with some universal constant 
$C(k)>0$ depending only on the order~$k$ of the integral and the 
quantities $\alpha_s$, defined in formula~(13.23) and~(13.24).}

\medskip
Let me finally  remark that there is a counterpart of Theorem~13.7
about degenerate $U$-statistics. Adamczak'w paper~[1] contains 
such a result. Here we do not discuss it, because this result is 
far from the main topic of this work. We only remark that some new 
quantities have to be introduced to formulate it. The appearance of 
these conditions is related to the fact that in an estimate about 
the tail-behaviour of a degenerate $U$-statistic we need a bound 
not only on the $L_2$-norm but also on the supremum norm of the 
kernel function. In a sharp estimate the bound about the supremum 
of the kernel function has to be replaced by a more complex system 
of conditions, just as the condition about the $L_2$-norm of the 
kernel function was replaced by a condition about the quantities 
$\alpha_s$, $1\le s\le k$, defined in formulas~(13.23) and~(13.24) 
in Theorem~13.7.

\beginsection 14. Reduction of the main result in this work.

The main result of this work is Theorem 8.4 or its multiple integral
version Theorem~8.2. It was shown in Section~9 that Theorem 8.2
follows from Theorems~8.4. Hence it is enough to prove Theorem~8.4.
It may be useful to study this problem together with its multiple
Wiener--It\^o integral version, Theorem~8.6.

Theorems~8.6 and~8.4 will be proved similarly to their one-variate
versions, Theorems~4.2 and~4.1. Theorem~8.6 will be proved with
the help of~Theorem~8.5 about the estimation of the tail
distribution of multiple Wiener--It\^o integrals. A natural 
modification of the chaining argument applied in the proof of
Theorem~4.2 works also in this case. No new difficulties arise. On 
the other hand, in the proof of Theorem~8.4 several new 
difficulties have to be overcome. I start with the proof of 
Theorem~8.6.

\medskip\noindent
{\it Proof of Theorem 8.6.}\/ Fix a number $0<\e<1$, and let us
list the elements of the countable set $\Cal F$ as $f_1,f_2,\dots$.
For all $p=0,1,2,\dots$ let us choose by exploiting the conditions
of Theorem~8.6 a set $\Cal F_p=\{f_{a(1,p)},\dots,f_{a(m_p,p)}\}
\subset\Cal F$ of function with
$m_p\le2D\,2^{(2p+4)L}\e^{-L}\sigma^{-L}$ elements in such a way
that $\inff_{1\le j\le m_p}\int(f-f_{a(j,p)})^2\,d\mu
\le 2^{-4p-8}\e^2\sigma^2$ for all $f\in\Cal F$, and beside this 
let $f_p\in\Cal F_p$. For all indices $a(j,p)$, $p=1,2,\dots$, 
$1\le j\le m_p$, choose a predecessor $a(j',p-1)$, $j'=j'(j,p)$, 
$1\le j'\le m_{p-1}$, in such a way that the functions 
$f_{a(j,p)}$ and $f_{a(j',p-1)}$ satisfy the relation
$\int|f_{a(j,p)}-f_{a(j',p-1)}|^2\,d\mu\le\e^2\sigma^22^{-4(p+1)}$.
Theorem~8.5 with the choice $\bar u=\bar u(p)=2^{-(p+1)}\e u$ and
$\bar\sigma=\bar\sigma(p)=2^{-2p-2}\e\sigma$ yields the estimates
$$
\aligned
P(A(j,p))&=P\(k!|Z_{\mu,k}(f_{a(j,p)}-f_{a(j',p-1)})|\ge
2^{-(1+p)}\e u\)\\
&\le C \exp\left\{-\frac12\(\frac{2^{p+1}u}\sigma\)^{2/k}\right\},
\qquad 1\le j\le m_p,
\endaligned \tag14.1
$$
for all $p=1,2,\dots$, and
$$
\aligned
P(B(s))&=P\(k!|Z_{\mu,k}(f_{a(0,s)})|\ge \(1-\frac \e2\)u\)\le
C\exp\left\{-\frac12\(\frac{\(1-\frac\e2\)u}\sigma\)^{2/k}\right\},\\
&\qquad 1\le s\le m_0.
\endaligned \tag14.2
$$
Since all $f\in\Cal F$ is the element of at least one set
$\Cal F_p$, $p=0,1,2,\dots$, ($f_p\in \Cal F_p$), the definition of
the predecessor of an index $a(j,p)$ and of the events $A(j,p)$ and
$B(s)$ in formulas (14.1) and (14.2) together with the previous
estimates imply that
$$  \allowdisplaybreaks
\align
&P\(\sup_{f\in\Cal F}k!|Z_{\mu,k}(f)|\ge u\)
\le P\(\bigcup_{p=1}^\infty\bigcup_{j=1}^{m_p}A(j,p)
\cup\bigcup_{s=1}^{m_0}B(s)\) \\
&\qquad \le \sum_{p=1}^\infty\sum_{j=1}^{m_p}P(A(j,p))
+\sum_{s=1}^{m_0}P(B(s)) \tag14.3 \\
&\qquad \le \sum_{p=1}^{\infty} 2CD2^{(2p+4)L}\e^{-L}\sigma^{-L}
\exp\left\{-\frac12\(\frac{2^{p+1}u}\sigma\)^{2/k} \right\}\\
&\qquad\qquad +2^{1+4L}CD\e^{-L}\sigma^{-L}
\exp\left\{-\frac12\(\frac{\(1-\frac\e2\)u}\sigma\)^{2/k}\right\}.
\endalign
$$
Standard calculation shows that if 
$u\ge ML^{k/2}\frac1\e\log^{k/2}\frac2\e\sigma\log^{k/2}\frac2\sigma$ 
with a sufficiently large constant~$M$, then the inequalities
$$
2^{(2p+4)L}\e^{-L}\sigma^{-L}
\exp\left\{-\frac12\(\frac{2^{p+1}u}\sigma\)^{2/k} \right\}\le
2^{-p}\left\{-\frac12\(\frac{(1-\e)u}\sigma\)^{2/k} \right\}
$$
hold for all $p=1,2\dots$, and
$$
2^{4L}\e^{-L}\sigma^{-L}
\exp\left\{-\frac12\(\frac{\(1-\frac\e2\)u}\sigma\)^{2/k}\right\}\le
\exp\left\{-\frac12\(\frac{\(1-\e\)u}\sigma\)^{2/k}\right\}.
$$

These inequalities together with relation (14.3) imply
relation (8.15). Theorem~8.6 is proved.

\medskip
The proof of Theorem~8.4 is harder. In this case the chaining
argument in itself does not supply the proof, since Theorem~8.3
gives a good estimate about the distribution of a degenerate
$U$-statistic only if it has a not too small variance. The same
difficulty appeared in the proof of Theorem~4.1, and the method
applied in that case will be adapted to the present situation.

A multivariate version of Proposition~6.1 will be proved in
Proposition~14.1, and another result which can be considered as 
a multidimensional version of Proposition~6.2 will be formulated 
in Proposition~14.2. It will be shown that Theorem~8.4 follows 
from Propositions~14.1 and~14.2. Most steps of these proofs can
be considered as a simple repetition of the corresponding 
arguments in the proof of the results in Section~6. Nevertheless, 
I wrote them down for the sake of completeness.

\medskip
The result formulated in Proposition~14.1 can be proved in almost 
the same way as its one-variate version, Proposition~6.1. The only 
essential difference is that now we have to apply a multivariate 
version of the Bernstein's inequality. In the calculations of the 
proof of Proposition~14.1 the term $(\frac u\sigma)^{2/k}$ shows a 
behaviour similar to the term $(\frac u\sigma)^2$ in Proposition~6.1.
Theorem~14.1 contains the  information we can get by applying 
Theorem~8.3 together with the chaining argument. Its main content, 
inequality~(14.4), yields a good estimate on the supremum of 
degenerated $U$-statistics if it is taken for an 
appropriate finite subclass $\Cal F_{\bar\sigma}$ of the original 
class of kernel functions~$\Cal F$. The class of kernel functions 
$\Cal F_{\bar\sigma}$ is a relatively dense subclass of $\Cal F$ in 
the $L_2$ norm. Proposition~14.1 also provides some useful estimates
on the value of the parameter~$\bar\sigma$ which describes how 
dense the class of functions $\Cal F_{\bar\sigma}$ is in $\Cal F$.

\medskip\noindent
{\bf Proposition 14.1.} {\it Let the $k$-fold power $(X^k,\Cal X^k)$ 
of a measurable space $(X,\Cal X)$ be given together with some 
probability measure $\mu$ on $(X,\Cal X)$ and such a countable
$L_2$-dense class $\Cal F$ of functions $f(x_1,\dots,x_k)$ of $k$
variables on $(X^k,\Cal X^k)$ with some exponent~$L\ge1$ and 
parameter $D\ge1$ with the following properties. All functions 
$f\in\Cal F$ are canonical with respect to the measure~$\mu$, and 
they satisfy conditions~(8.4) and~(8.5) with some real number 
$0<\sigma\le1$. Take a sequence of independent $\mu$-distributed 
random variables $\xi_1,\dots,\xi_n$, $n\ge\max(k,2)$, and consider 
the (degenerate) $U$-statistics $I_{n,k}(f)$, $f\in \Cal F$, defined 
in formula (8.7), and fix some number $\bar A=\bar A_k\ge2^k$.

There is a number $M=M(\bar A,k)$ such that for all numbers~$u>0$ 
for which the inequality
$n\sigma^2\ge\(\frac u\sigma\)^{2/k}\ge M(L\log\frac2\sigma+\log D)$ 
holds, a number $\bar\sigma=\bar\sigma(u)$, 
$0\le\bar\sigma\le \sigma\le1$, 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
$\Cal F=\bigcupp_{j=1}^m\Cal D_j$, and for the (degenerate) 
$U$-statistics $I_{n,k}(f)$, $f\in\Cal F_{\bar\sigma(u)}$ the 
inequality
 $$ \allowdisplaybreaks
\align
P&\(\sup_{f\in\Cal F_{\bar\sigma(u)}}n^{-k/2}|I_{n,k}(f)|\ge \frac
u{\bar A}\)\le 2C\exp\left\{-\alpha\(\frac u{10\bar
A\sigma}\)^{2/k}\right\}  \tag14.4 \\
&\qquad \qquad \text{if}\quad n\sigma^2\ge \(\frac u\sigma\)^{2/k}
\ge M(L\log\frac2\sigma+\log D)
\endalign
$$
holds with the constants $\alpha=\alpha(k)$, $C=C(k)$ appearing in
formula~($8.10'$) of the Corollary of Theorem~8.3 and the exponent
$L$ and parameter $D$ of the $L_2$-dense class $\Cal F$. Beside 
this also the inequality $4\(\frac u{\bar A\bar\sigma}\)^{2/k}\ge
n\bar\sigma^2\ge\frac1{64}\(\frac u{\bar A\sigma}\)^{2/k}$ 
holds for this number $\bar\sigma=\bar\sigma(u)$. If the
number~$u$ satisfies also the inequality 
$$
n\sigma^2\ge \(\frac u\sigma\)^{2/k}\ge
M(L^{3/2}\log\frac2\sigma +(\log D)^{3/2}) \tag14.5
$$ 
with a sufficiently large number $M=M(\bar A,k)$, then the relation
$n\bar\sigma^2\ge L\log n+\log D$ holds, too.}

\medskip\noindent
{\it Proof of Proposition 14.1.} Let us list the elements of the
countable set $\Cal F$ as $f_1,f_2,\dots$. For all $p=0,1,2,\dots$
let us choose, by exploiting the $L_2$-density property of the class
$\Cal F$, a set $\Cal F_p=\{f_{a(1,p)},\dots,f_{a(m_p,p)}\}\subset
\Cal F$ with $m_p\le D\,2^{2pL}\sigma^{-L}$ elements in such a way
that $\inff_{1\le j\le m_p}\int (f-f_{a(j,p)})^2\,d\mu\le
2^{-4p}\sigma^2$ for all $f\in\Cal F$.
For all indices $a(j,p)$, $p=1,2,\dots$, $1\le j\le m_p$, choose a
predecessor $a(j',p-1)$, $j'=j'(j,p)$, $1\le j'\le m_{p-1}$, in such a
way that the functions $f_{a(j,p)}$ and $f_{a(j',p-1)}$ satisfy the
relation $\int|f_{a(j,p)}-f_{a(j',p-1)}|^2\,d\mu\le \sigma^2
2^{-4(p-1)}$. Then the inequalities
$\int\(\frac{f_{a(j,p)}-f_{a(j',p-1)}}2\)^2\,d\mu\le4\sigma^2 2^{-4p}$
and $\supp_{x_j\in X,\,1\le j\le k}\left|
\frac{f_{a(j,p)}(x_1,\dots,x_k)-f_{a(j',p-1)}(x_1,\dots,x_k)}2\right|
\le 1$ hold. The Corollary of Theorem~8.3  yields that
$$
\aligned
P(A(j,p))&=P\(n^{-k/2}|I_{n,k}(f_{a(j,p)}-f_{a(j',p-1)})|\ge
\frac{2^{-(1+p)}u}{\bar A}\)\\
&\le C \exp\left\{-\alpha\(\frac{2^{p}u}{8\bar A
\sigma}\)^{2/k} \right\}
\quad \text {if}\quad 4n\sigma^2 2^{-4p}\ge\(\frac{2^{p}u}
{8\bar A\sigma}\)^{2/k}, \\
&\qquad\qquad 1\le j\le m_p,\; p=1,2,\dots,
\endaligned \tag14.6
$$
and
$$
\align
P(B(s))&=P\(n^{-k/2}|I_{n,k}(f_{0,s})|\ge \frac u{2\bar A}\)\le
C\exp\left\{-\alpha\(\frac u{2\bar A\sigma}\)^{2/k}\right\},
\quad 1\le s\le m_0,   \\
&\qquad\qquad\qquad\text{if} \quad n\sigma^2\ge \(\frac u{2\bar
A\sigma}\)^{2/k}.  \tag14.7
\endalign
$$
Introduce an integer $R=R(u)$, $R>0$, which satisfies the relations
$$
2^{(4+{2/k})(R+1)}\(\frac{u}{\bar A\sigma}\)^{2/k} \ge
2^{2+6/k}n\sigma^2\ge 2^{(4+2/k)R}\(\frac{u}{\bar A\sigma}\)^{2/k},
$$
and define $\bar\sigma^2=2^{-4R}\sigma^2$ and $\Cal
F_{\bar\sigma}=\Cal F_R$ (this is the class of functions $\Cal F_p$
introduced at the start of the proof with $p=R$). We defined the
number~$R$, analogously to the proof of Theorem~6.1, as the largest
number~$p$ for which the condition formulated in~(14.6) holds. As
$n\sigma^2\ge\(\frac u\sigma\)^{2/k}$, and $\bar A\ge2^k$ by our
conditions, there exists such a positive integer $R$.) 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 the inequalities
(14.6) and (14.7) hold for $1\le p\le R$. Hence the
definition of the predecessor of an index $a(j,p)$ implies that
$$
\align
&P\(\sup_{f\in\Cal F_{\bar\sigma}}n^{-k/2}|I_{n,k}(f)|\ge \frac u{\bar
A}\) \le P\(\bigcup_{p=1}^R\bigcup_{j=1}^{m_p}A(j,p)
\cup\bigcup_{s=1}^{m_0}B(s)\) \\
&\qquad \le \sum_{p=1}^R\sum_{j=1}^{m_p}P(A(j,p))+\sum_{s=1}^{m_0}P(B(s))
\le \sum_{p=1}^{\infty} CD\,2^{2pL}\sigma^{-L}
\exp\left\{-\alpha\(\frac{2^{p}u}{8\bar A\sigma}\)^{2/k}
\right\}\\
&\qquad\qquad +CD\sigma^{-L}\exp\left\{-\alpha\(\frac
u{2\bar A\sigma}\)^{2/k}\right\}.
\endalign
$$
If the condition $\(\frac u\sigma\)^{2/k}\ge 
M(L\log\frac2\sigma+\log D)$ holds with a sufficiently large 
constant $M$ (depending on $\bar A$), then the inequalities
$$
D2^{2pL}\sigma^{-L}\exp\left\{-\alpha\(\frac{2^{p}u}{8\bar
A\sigma}\)^{2/k} \right\}\le 2^{-p}\exp\left\{-\alpha\(\frac{2^{p}u}
{10\bar A \sigma}\)^{2/k} \right\}
$$
hold for all $p=1,2,\dots$, and
$$
D\sigma^{-L}\exp\left\{-\alpha\(\frac u{2\bar A\sigma}\)^{2/k}\right\}
\le\exp\left\{-\alpha\(\frac u{10\bar A\sigma}\)^{2/k}\right\}.
$$
Hence the previous estimate implies that
$$
\align
&P\(\sup_{f\in\Cal F_{\bar\sigma}}n^{-k/2}|I_{n,k}(f)|\ge
\frac u{\bar A}\) \le\sum_{p=1}^{\infty}C 2^{-p}
\exp\left\{-\alpha\(\frac{2^{p}u}{10\bar A \sigma}\)^{2/k}
\right\}\\
&\qquad +C\exp\left\{-\alpha\(\frac u{10\bar A
\sigma}\)^{2/k}\right\} \le 2C\exp\left\{-\alpha
\(\frac u{10 \bar A\sigma}\)^{2/k}\right\},
\endalign
$$
and relation (14.4) holds.

The estimates
$$
\align
\frac1{64}\(\frac u{\bar A\sigma}\)^{2/k}
&\le2^{-2-6/k}2^{2k/R}\(\frac u{\bar A\sigma}\)^{2/k}
=2^{-4R}\cdot2^{(4+2/k)R-2-6/k}\(\frac{u}{\bar A\sigma}\)^{2/k}\\
&\le n\bar\sigma^2=2^{-4R} n\sigma^2\le
2^{-4R}\cdot2^{(4+2/k)(R+1)-2-6/k}\(\frac{u}{\bar A\sigma}\)^{2/k}\\
&=2^{2-4/k}\cdot 2^{2R/k}\(\frac{u}{\bar A \sigma}\)^{2/k}
=2^{2-4/k}\cdot2^{-2R/k} \(\frac{u}{\bar A\bar\sigma}\)^{2/k}
\le 4 \(\frac{u}{\bar A\bar\sigma}\)^{2/k}
\endalign
$$
hold because of the relation~$R\ge1$. This means that $n\bar\sigma^2$
has the upper and lower bound formulated in Proposition~14.1. 
It remained to show that $n\bar\sigma^2\ge
L\log n+D$ if relation~(14.5) holds.

This inequality clearly holds under the conditions of Proposition~14.1
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}\(\frac u {\bar
A\sigma}\)^{2/k} \ge\frac1{64}\bar A^{-2/k}
M(L^{3/2}\log\frac2\sigma +(\log D)^{3/2})^{3/2}\ge
\frac1{192}\bar A^{-2/k} M(L^{3/2}\log n+(\log D)^{3/2})
\ge L\log n+\log D$ if $M= M(\bar A,k)$ is sufficiently large.

If $\sigma\ge n^{-1/3}$, then the inequality $2^{(4+2/k)R}\(\frac
u{\bar A\sigma}\)^{2/k} \le2^{2+6/k} n\sigma^2$ can be applied. This
implies that $2^{-4R}\ge2^{-4(2+6/k))/(4+2/k)}\left[\dfrac{\(\frac
u{\bar A\sigma}\)^{2/k}}{n\sigma^2}\right]^{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}\[\(\frac u\sigma\)^{2/k}\]^\gamma
\text{ with } \gamma=\frac4{4+\frac2k}\ge\frac23.
$$
The inequalities $n\sigma^2\ge n^{1/3}$ and 
$n\sigma^2\ge(\frac u\sigma)^{2/k}
\ge M(L^{3/2}\log\frac2\sigma+(\log D)^{3/2})
\ge\frac M2(L^{3/2}+(\log D)^{3/2})$ hold, 
(since $\log\frac2\sigma\ge\frac12$). They yield that for 
sufficiently large $M=M(\bar A,k)$
$(n\sigma^2)^{1-\gamma}\[\(\frac u\sigma\)^{2/k}\]^\gamma\ge
(n\sigma^2)^{1-\gamma}\[\(\frac u\sigma\)^{2/k}\]^{2/3}=
(n\sigma^2)^{1/(2k+1)}\[\(\frac u\sigma\)^{2/k}\]^{2/3}$, and
$$
\align
n\bar\sigma^2&\ge \frac{\bar A^{-4/3}}{50} (n\sigma^2)^{1/(2k+1)}\[\(\frac
u\sigma\)^{2/k}\]^{2/3}\\
&\ge \frac{\bar A^{-4/3}}{50}n^{1/3(2k+1)}
\(\frac M2\)^{2/3} (L^{3/2}+(\log D)^{3/2})^{2/3} \ge L\log n+\log D.
\endalign
$$


\medskip
A multivariate analog of Proposition~6.2 is formulated in
Proposition~14.2, and it will be shown that Propositions~14.1
and~14.2 imply Theorem~8.4.

\medskip\noindent
{\bf Proposition 14.2.} {\it Let a probability measure $\mu$ be
given on a measurable space $(X,\Cal X)$ together with 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 (with respect to the measure~$\mu$) kernel functions
$f=f(x_1,\dots,x_k)$ with some parameter $D\ge1$ and exponent 
$L\ge1$ on the product space $(X^k,\Cal X^k)$. Let all functions 
$f\in\Cal F$ satisfy conditions~(8.1) and~(8.2) with some 
$0<\sigma\le1$ such that $n\sigma^2>L\log n+D$. Let us consider 
the (degenerate) $U$-statistics $I_{n,k}(f)$ with the random 
sequence $\xi_1,\dots,\xi_n$, $n\ge\max(2,k)$, and kernel 
functions $f\in\Cal F$. There exists a threshold index 
$A_0=A_0(k)>0$ and two numbers $\bar C=\bar C(k)>0$ and 
$\gamma=\gamma(k)>0$ depending only on the order $k$ of the 
$U$-statistics such that the degenerate $U$-statistics 
$I_{n,k}(f)$, $f\in\Cal F$, 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.
\tag14.8
$$
}

\medskip
Proposition~14.2 yields an estimate for the tail distribution of
the supremum of degenerate $U$-statistics at level
$u\ge A_0n^{k/2}\sigma^{k+1}$, i.e. in the case when Theorem~8.3
does not give a good estimate on the tail-distribution of the single
degenerate $U$-statistics taking part in the supremum at the left-hand
side of~(14.8).

Formula (8.11) will be proved by means of Proposition~14.1 with an
appropriate choice of the parameter $\bar A$ in it and 
Proposition~14.2 with the choice $\sigma=\bar\sigma=\bar\sigma(u)$ 
and the classes of functions
$\Cal F_j=\left\{\frac{g-f_j}2\colon\; g\in\Cal D_j\right\}$ 
with the number $\bar\sigma$, functions~$f_j$ and sets of 
functions~$\Cal D_j$, $1\le j\le m$, introduced in Proposition~14.1. 
Clearly,
$$
\aligned
&P\(\supp_{f\in\Cal F}n^{-k/2}|I_{n,k}(f)|\ge u\)\le
P\(\sup_{f\in\Cal F_{\bar\sigma}}n^{-k/2}|I_{n,k}(f)|\ge \frac u{\bar
A}\) \\
&\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\(\frac12-\frac1{2\bar A}\)u\),
\endaligned \tag14.9
$$
where $m$ is the cardinality of the set of functions $\Cal
F_{\bar\sigma}$ appearing in Proposition~14.1. 
We shall estimate the two terms of the sum at the right-hand side
of~(14.9) by means of Propositions~14.1 and~14.2 with a good choice
of the parameters $\bar A$ and the corresponding $M=M(\bar A)$ in
Proposition~14.1, and the parameter $\bar A_0=A\ge A_0$ in 
Proposition~14.2.

We shall choose the parameter~$\bar A_0\ge A_0$ so that it satisfies
also the relation $\gamma\bar A_0^{1/2k}\ge2$ with the 
number~$\gamma$ appearing in Proposition~14.2, hence we put
$\bar A_0=\max(A_0,(\frac2\gamma)^{2k})$. We want to define the
parameter $\bar A$ in such a way that the numbers~$u$ satisfying the
conditions of Proposition~14.1 also satisfy the relation 
$(\frac12-\frac1{2\bar A})u\ge\bar A_0n^{k/2}\bar\sigma^{k+1}$ with
the already fixed number~$\bar A_0$. This inequality can be rewritten
in the form $\bar A_0^{-2/k}(\frac12-\frac1{2\bar A})^{2/k}
(\frac u{\bar\sigma})^{2/k}\ge n\bar\sigma^2$. On the other hand, we
have under the conditions of Proposition~14.1 
$4(\frac u{\bar A\bar\sigma})^{2/k}\ge n\bar\sigma^2$. Hence the
desired inequality holds if 
$\bar A_0^{-2/k}(\frac12-\frac1{2\bar A})^{2/k}\ge 4{\bar A}^{-2/k}$,
thus it holds with the choice $\bar A=2^{k+1}\bar A_0+1$.

With such a choice of $\bar A$ (together with the corresponding 
$M=M(\bar A,k)$) and $\bar A_0$ we can write 
$$
\align
&P\(\sup_{g\in\Cal D_j} n^{-k/2}
\left|I_{n,k}\(\frac{f_j-g}2\)\right| \ge\(\frac12-\frac1{2\bar
A}\)u\)\\
&\qquad\le P\(\sup_{g\in\Cal D_j}n^{-k/2}
\left|I_{n,k}\(\frac{f_j-g}2\)\right|
\ge\bar A_0n^{k/2}\bar\sigma^{k+1}\)
\le \bar Ce^{-\gamma\bar A_0^{1/2k}n\bar\sigma^2}
\endalign
$$
for all $1\le j\le m$.
(Observe that the set of functions $\frac{f_j-g}2,\;g\in\Cal D_j$, is
an $L_2$-dense class with parameter $D$ and exponent $L$.) Hence
Proposition~14.1 (relation (14.4) together with the inequality $m\le
D\bar \sigma^{-L}$) and formula (14.8) with $A=\bar A_0$ and 
relation~(14.9) imply that
$$
P\(\supp_{f\in\Cal F} n^{-k/2}|I_{n,k}(f)|\ge u\)
\le 2C\exp\left\{-\alpha\(\frac u{10\bar A\sigma}\)^{2/k}\right\}
+\bar CD\bar\sigma^{-L} e^{-\gamma\bar A_0^{1/2k}n\bar\sigma^2}.
\tag14.10
$$
We show by repeating the corresponding argument in Section~6 that
$D\bar\sigma^{-L}\le e^{n\bar\sigma^2}$. Indeed, we have to show 
that $\log D+L\log\frac1{\bar\sigma}\le n\bar\sigma^2$. But, as we 
have seen, the relation $n\bar\sigma^2\ge L\log n+\log D$ with 
$L\ge1$ and $D\ge1$ implies that $n\bar\sigma^2\ge\log n$, hence 
$\log\frac1\sigma\le\log n$, and 
$\log D+L\log\frac1{\bar\sigma}\le \log D+L\log n\le n\bar\sigma^2$.
On the other hand, $\gamma\bar A_0^{1/2k}\ge2$ by the definition of
the number~$\bar A_0$, and by the estimates of Proposition~14.1
$n\bar\sigma^2\ge\frac1{64}\(\frac u{\bar A\sigma}\)^{2/k}$. The
above relations imply that $D\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 (14.10) yields that
$$
\align
&P\(\supp_{f\in\Cal F}n^{-k/2}|I_{n,k}(f)|\ge u\)\\
&\qquad\le 2C\exp \left\{-\frac\alpha{(10\bar A)^2}\(\frac
u\sigma\)^{2/k}\right\} +\bar C\exp\left\{-\frac\gamma{128}
\bar A_0^{1/2k} \bar A^{-2/k} \(\frac u\sigma\)^{2/k}\right\},
\endalign
$$
and this estimate implies Theorem~8.4.

\medskip
To complete the proof of Theorem~8.4 we have to prove
Proposition~14.2. It will be proved, similarly to its one-variate
version Proposition~6.2, by means of a symmetrization argument. 
We want to find its right formulation. It would be natural to 
formulate it as a result about the supremum of degenerate 
$U$-statistics. However, we shall choose a slightly different 
approach. There is a notion, called decoupled $U$-statistic. 
Decoupled $U$-statistics behave similarly to $U$-statistics, but 
it is simpler to work with them, because they have more 
independence properties. It turned out to be useful to introduce 
this notion and to apply a result of de la Pe\~na and 
Montgomery--Smith which enables us to reduce the estimation of 
$U$-statistics to the estimation of decoupled $U$-statistics, 
and to work out the symmetrization argument for decoupled 
$U$-statistics.

Next we  introduce the notion of decoupled $U$-statistics 
together with their randomized version. We also formulate a 
result of de la Pe\~na and Montgomery--Smith in Theorem~14.3 
which enables us to reduce Proposition~14.2 to a version of it, 
presented in Proposition~$14.2'$. It states a result similar 
to Proposition~14.2 about decoupled $U$-statistics. The proof of 
Proposition~$14.2'$ is the hardest part of the problem. In 
Sections~15, 16 and~17 we deal essentially with this problem.
The result of de la Pe\~na and Montgomery--Smith will be 
proved in Appendix~D.

Now we introduce the following notions.

\medskip\noindent
{\bf The definition of decoupled and randomized
decoupled $U$-statistics.} {\it Let us have $k$ independent
copies $\xi_1^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$, of a sequence
$\xi_1,\dots,\xi_n$ of independent and identically distributed
random variables taking their values in a measurable space
$(X,\Cal X)$ together with a measurable function $f(x_1,\dots,x_k)$
on the product space $(X^k,\Cal X^k)$ with values in a separable
Banach space. The decoupled $U$-statistic $\bar I_{n,k}(f)$
determined by the random sequences $\xi_1^{(j)},\dots,\xi_n^{(j)}$,
$1\le j\le k$, and kernel function $f$ is defined by the formula
$$
\bar I_{n,k}(f)=\frac1{k!}\summ\Sb (l_1,\dots,l_k)\colon \;
1\le l_j\le n,\;j=1,\dots,k,\\
l_j\neq l_{j'} \text{ if } j\neq j'\endSb
f\(\xi_{l_1}^{(1)},\dots,\xi_{l_k}^{(k)}\). \tag14.11
$$

Let us have beside the sequences
$\xi_1^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$, and function
$f(x_1,\dots,x_k)$ a sequence of independent random variables
$\e=(\e_1,\dots,\e_n)$, $P(\e_l=1)=P(\e_l=-1)=\frac12$,
$1\le l\le n$, which is independent also of the sequences of
random variables $\xi_1^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$.
The randomized decoupled $U$-statistic $\bar I_{n,k}(f,\e)$
(depending on the random sequences $\xi_1^{(j)},\dots,\xi_n^{(j)}$,
$1\le j\le k$, the kernel function $f$ and the randomizing sequence
$\e_1,\dots,\e_n$) is defined by the formula
$$
\bar I^\e_{n,k}(f)=\frac1{k!}\summ\Sb (l_1,\dots,l_k)\colon \;
1\le l_j\le n,\;j=1,\dots,k,\\
l_j\neq l_{j'} \text{ if } j\neq j'\endSb
\e_{l_1}\cdots\e_{l_k}f\(\xi_{l_1}^{(1)},\dots,\xi_{l_k}^{(k)}\).
\tag14.12
$$
}

\medskip
A decoupled or randomized decoupled $U$-statistics (with real
valued kernel function) will be called degenerate if its kernel
function is canonical. This terminology is in full accordance with
the definition of (usual) degenerate $U$-statistics.

A result of de la Pe\~na and Montgomery--Smith will be formulated
below. It gives an upper bound for the tail distribution of a
$U$-statistic by means of the tail distribution of an appropriate
decoupled $U$-statistic. It also has a generalization, where the
supremum of $U$-statistics is bounded by the supremum of decoupled
$U$-statistics. It enables us to reduce Proposition~14.2 to a 
version formulated Proposition~$14.2'$, which gives a bound on the 
tail distribution of the supremum of decoupled $U$-statistics. 
It is simpler to prove this result than the original one.

Before the formulation of the theorem of de la Pe\~na and
Montgomery--Smith I make some remark about it. It considers
more general $U$-statistics with kernel functions taking values
in a separable Banach space, and it compares the norm of
Banach space valued $U$-statistics and decoupled $U$-statistics.
(Decoupled $U$-statistics were defined with general Banach space
valued kernel functions, and the definition of $U$-statistics can
also be generalized to separable Banach space valued kernel
functions in a natural way.) This result was formulated in such
a general form for a special reason. This helped to derive 
formula~(14.14) of the subsequent theorem from formula~(14.13). 
It can be exploited in the proof of formula~(14.14) that the 
constants in the estimate~(14.13) do not depend on the Banach 
space, where the kernel function~$f$ takes its values. 

\medskip\noindent
{\bf Theorem 14.3. (Theorem of de la Pe\~na and Montgomery--Smith
about the comparison of $U$-statistics and decoupled
$U$-statistics).} {\it Let us consider a sequence of independent
and identically distributed random variables $\xi_1,\dots,\xi_n$
with values in a measurable space $(X,\Cal X)$ together with $k$ 
independent copies $\xi_{1}^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$, 
of this sequence. Let us also have a function $f(x_1,\dots,x_k)$ on 
the $k$-fold product space $(X^k,\Cal X^k)$ which takes its values 
in a separable Banach space~$B$. Let us take the $U$-statistic and
decoupled $U$-statistic $I_{n,k}(f)$ and $\bar I_{n,k}(f)$ with
the help of the above random sequences $\xi_1,\dots,\xi_n$,
$\xi_{1}^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$, and kernel
function~$f$. There exist some constants $\bar C=\bar C(k)>0$
and $\gamma=\gamma(k)>0$ depending only on the order~$k$ of the
$U$-statistic such that
$$
P\(\|I_{n,k}(f)\|>u\)\le\bar CP\(\|\bar I_{n,k}(f)\|>\gamma u\)
\tag14.13
$$
for all $u>0$. Here $\|\cdot\|$ denotes the norm in the Banach
space~$B$ where the function~$f$ takes its values.

More generally, if we have a countable sequence of functions
$f_s$, $s=1,2,\dots$, taking their values in the same separable
Banach-space, then
$$
P\(\sup_{1\le s<\infty} \left\| I_{n,k}(f_s)\right\|>u\)\le
\bar CP\(\sup_{1\le s<\infty}\left\|\bar I_{n,k}(f_s)\right\|
>\gamma u\). \tag14.14
$$
}
\medskip
Now I formulate the following version of Proposition~4.2.

\medskip\noindent
{\bf Proposition $14.2'$.} {\it Let a probability measure $\mu$ be
given on a measurable space $(X,\Cal X)$ together with a sequence
of independent and $\mu$ distributed random variables
$\xi_1,\dots,\xi_n$, $n\ge\max(k,2)$, and a countable $L_2$-dense 
class $\Cal F$ of canonical (with respect to the measure~$\mu$) 
kernel functions $f=f(x_1,\dots,x_k)$ with some parameter $D\ge1$ 
and exponent $L\ge1$ on the product space $(X^k,\Cal X^k)$. Let 
all functions $f\in\Cal F$ satisfy conditions~(8.1) and~(8.2) 
with some $0<\sigma\le1$ such that $n\sigma^2>L\log n+\log D$.
Let us take $k$ independent copies $\xi_1^{(j)},\dots,\xi_n^{(j)}$, 
$1\le j\le k$, of the random sequence $\xi_1,\dots,\xi_n$, and 
consider the decoupled $U$-statistics $\bar I_{n,k}(f)$, 
$f\in\Cal F$, defined with their help in formula~(14.11).

There exists a threshold index $A_0=A_0(k)>0$ depending only on 
the order $k$ of the decoupled $U$-statistics $I_{n,k}(f)$, 
$f\in\Cal F$, such that the (degenerate) decoupled 
$U$-statistics $\bar I_{n,k}(f)$, $f\in\Cal F$, satisfy the 
following version of inequality (14.8):
$$
P\(\sup_{f\in\Cal F}n^{-k/2}|\bar I_{n,k}(f)|
\ge An^{k/2}\sigma^{k+1}\)
\le e^{-2^{-(1/2+1/2k)} A^{1/2k}n\sigma^2}\quad \text{if } A\ge A_0.
\tag14.15
$$
}

\medskip
It is clear that Proposition~$14.2'$ and Theorem~14.3, more
explicitly formula~(14.14) in it, imply Proposition 14.2. Hence the
proof of Theorem~8.4 was reduced to Proposition~$14.2'$ in this
section. The proof of Proposition~$14.2'$ is based on a symmetrization
argument. Its main ideas will be explained in the next section.

\beginsection 15. The strategy of the proof for the main result of
this work.

In the previous section the proof of Theorem~8.4 was reduced to
that of Proposition~$14.2'$. Proposition~$14.2'$ is a multivariate
version of Proposition~6.2, and its proof is based on similar
ideas. An important step in the proof of Theorem~6.2 was a 
symmetrization argument in which we reduced the estimation of
the probability $P\(\supp_{f\in\Cal F}\summ_{j=1}^nf(\xi_j)>u\)$
to that of the probability
$P\(\supp_{f\in\Cal F}\summ_{j=1}^n\e_jf(\xi_j)>\frac u3\)$, where
$\xi_1,\dots,\xi_n$ is a sequence of independent and identically 
distributed random variables, and $\e_j$, $1\le j\le n$, is a 
sequence of independent random variables with distribution 
$P(\e_j=1)=P(\e_j=-1)=\frac12$, independent of the 
sequence~$\xi_j$. Let us understand how to formulate the 
corresponding symmetrization argument in the proof of 
Proposition~$14.2'$ and how to prove it.

The symmetrization argument applied in the proof of Proposition~6.2
was based on two observations. We took a copy $\xi_1',\dots,\xi'_n$ 
of the sequence $\xi_1,\dots,\xi_n$, i.e. a sequence of independent 
random variables which is independent also of the original 
sequence $\xi_1,\dots,\xi_n$, and has the same distribution. In the 
first step we compared the tail distribution of the expression 
$\supp_{f\in\Cal F}\summ_{j=1}^n[f(\xi_j)-f(\xi'_j)]$
with that of $\supp_{f\in\Cal F}\summ_{j=1}^nf(\xi_j)$. This
was done with the help of Lemma~7.1. In the second step, in
Lemma~7.2, we proved a `randomization argument' which stated that 
the distribution of the random fields 
$\summ_{j=1}^n[f(\xi_j)-f(\xi_j')]$ and 
$\summ_{j=1}^n\e_j[f(\xi_j)-f(\xi_j')]$, $f\in\Cal F$, agree.
The symmetrization argument was proved with the help of these
two observations.

In the proof of Proposition~$14.2'$ we would like to reduce the 
estimation of the tail distribution of the supremum of decoupled 
$U$-statistics $\supp_{f\in\Cal F}\bar I_{n,k}(f)$ defined in
formula~(14.11) to the estimation of the tail distribution of
the supremum of randomized decoupled $U$-statistics 
$\supp_{f\in\Cal F}\bar I_{n,k}^\e(f)$ defined in formula~(14.12)
by means of a similar argument. To do this first we have to 
understand what kind of random fields should be introduced 
instead of $\summ_{j=1}^n[f(\xi_j)-f(\xi'_j)]$, $f\in\Cal F$, in
the new case. In formula~(15.1) we shall define such a random 
field. Its definition reminds a bit to the definition of 
Stieltjes measures. In Lemma~15.1 we will show that a version
of the `randomization argument' of Lemma~7.2 can be applied when
we working with this random field.

The adaptation of the first step of the symmetrization argument 
in the proof of Proposition~6.2 to the present case is much harder. 
The proof of Proposition~6.2 was based on the symmetrization lemma, 
Lemma~7.1, which does not work in the present case. Hence we shall 
prove a generalization of this result in Lemma~15.2. The proof is 
difficult even with the help of this result. The hardest part of 
the problem appears at this point. I shall write about it after 
the formulation of Lemma~15.2.

To formulate Lemma~15.1 needed in our proof we introduce some
notations.

Let $\Cal V_k$ denote the set of all sequences $(v(1),\dots,v(k))$
of length~$k$ such that $v(j)=+1$ or $v(j)=-1$ for all $1\le j\le k$.
Let $m(v)$, $v=(v(1),\dots,v(k))\in\Cal V_k$, denote the number of
digits $-1$ in the sequence $v$. Let a (real valued) function
$f(x_1,\dots,x_k)$ of $k$ variables be given on a measurable space
$(X,\Cal X)$ together with a sequence of independent and identically
distributed random variables $\xi_1,\dots,\xi_n$ with values in the
space $(X,\Cal X)$ and $2k$ independent copies
$\xi_1^{(j,1)},\dots,\xi_n^{(j,1)}$ and
$\xi_1^{(j,-1)},\dots,\xi_n^{(j,-1)}$, $1\le j\le k$, of this
sequence. Let us have beside them another sequence
$\e=(\e_1,\dots,\e_n)$, $P(\e_j=1)=P(\e_j=-1)=\frac12$, of independent
random variables, also independent of all previously introduced random
variables. With the help of the above quantities we introduce the 
random variables
$$
\tilde I_{n,k}(f)=\frac1{k!}\sum_{v\in \Cal V_k}
(-1)^{m(v)} \summ\Sb (l_1,\dots,l_k)\colon\; 1\le l_r\le n,\;
r=1,\dots, k,\\
l_r\neq l_{r'} \text{ if } r\neq r'\endSb
f\(\xi_{l_1}^{(1,v(1))},\dots,\xi_{l_k}^{(k,v(k))}\) \tag15.1
$$
and
$$
\tilde I^\e_{n,k}(f)=\frac1{k!}\sum_{v\in \Cal V_k}
(-1)^{m(v)} \summ\Sb (l_1,\dots,l_k)\colon\; 1\le l_r\le n,\;
r=1,\dots, k,\\
l_r\neq l_{r'} \text{ if } r\neq r'\endSb \e_{l_1}\cdots\e_{l_k}
f\(\xi_{l_1}^{(1,v(1))},\dots,\xi_{l_k}^{(k,v(k))}\) \tag15.2
$$
The number $m(v)$ in the above formulas denotes the number of the 
digits $-1$ in the $\pm1$ sequence $v$ of length~$k$, hence it 
counts how many random variables $\xi_{l_j}^{(j,1)}$, $1\le j\le k$, 
were replaced by the `secondary copy' $\xi_{l_j}^{(j,-1)}$ for a 
$v\in\Cal V_k$ in the inner sum in formulas~(15.1) or~(15.2).

The following result holds.

\medskip\noindent
{\bf Lemma 15.1.} {\it Let us consider a (non-empty) class of
functions $\Cal F$ of $k$ variables $f(x_1,\dots,x_k)$ on the
space $(X^k,\Cal X^k)$ together with the random variables
$\tilde I_{n,k}(f)$ and $\tilde I^\e_{n,k}(f)$ defined in
formulas~(15.1) and (15.2) for all $f\in \Cal F$. The 
distributions of the random fields $\tilde I_{n,k}(f)$, 
$f\in\Cal F$, and $\tilde I^\e_{n,k}(f)$, $f\in \Cal F$, agree.}

\medskip
Let me recall that we say that the distribution of two random
fields $X(f)$, $f\in\Cal F$, and $Y(f)$, $f\in\Cal F$, 
agree if for any finite sets $\{f_1,\dots,f_p\}\in\Cal F$ the 
distribution of the random vectors $(X(f_1),\dots,X(f_p))$ and
$(Y(f_1),\dots,Y(f_p))$ agree. 
%Formulas~(15.1) and~(15.2) show  some similarity with the 
%formula by which Stieltjes measures are defined in the 
%$k$-dimensional space by means of a functions of $k$~variables.

\medskip\noindent
{\it Proof of Lemma 15.1.}\/ I even claim that for any fixed
sequence $u=(u(1),\dots,u(n))$, $u(l)=\pm1$, $1\le l\le n$,
of length~$n$, the conditional distribution of the field 
$\tilde I^\e_{n,k}(f)$, $f\in \Cal F$, under the condition 
$(\e_1,\dots,\e_n)=u=(u(1),\dots,u(n))$ agrees with the 
distribution of the field of $\tilde I_{n,k}(f)$, $f\in\Cal F$.

Indeed, the random variables $\tilde I_{n,k}(f)$, $f\in\Cal F$, 
defined in (15.1) are functions of a random vector with coordinates
$(\xi_l^{(j)},\bar\xi_l^{(j)})=(\xi^{(j,1)}_l,\xi^{(j,-1)}_l)$,
$1\le l\le n$, $1\le j\le k$, and the distribution of this random
vector does not change if the coordinates 
$(\xi_{l}^{(j)},\bar\xi_l^{(j)})=(\xi^{(j,1)}_l,\xi^{(j,-1)}_l)$
with such indices $(l,j)$ for which $u(l)=-1$ (and the index~$j$ 
is arbitrary) are replaced by
$(\bar\xi_l^{(j)},\xi_l^{(j)})=(\xi^{(j,-1)}_l,\xi^{(j,1)}_l)$,
and the coordinates $(\xi_{l}^{(j)},\bar\xi_l^{(j)})$ with such 
indices $(l,j)$ for which $u(l)=1$ are not changed. As a 
consequence, the distribution of the random field 
$\tilde I_{n,k}(f|u)$, $f\in\Cal F$, we get by replacing the 
original vector $(\xi_l^{(j)},\bar\xi_l^{(j)})$, $1\le l\le n$, 
$1\le j\le k$, in the definition of the expression 
$\tilde I_{n,k}(f)$ in~(15.1) for all $f\in \Cal F$ by this 
modified vector depending on~$u$ has the same distribution as the 
random field $\tilde I_{n,k}(f)$, $f\in\Cal F$. On the other hand, 
I claim that the distribution of the random field 
$\tilde I_{n,k}(f|u)$, $f\in\Cal F$, agrees with the conditional 
distribution of the random field $\tilde I^\e_{n,k}(f)$, 
$f\in\Cal F$, defined in~(15.2) under the condition that 
$(\e_1,\dots,\e_n)=u$ with $u=(u(1),\dots,u(n))$.

To prove the last statement let us observe that the conditional
distribution of the random field $\tilde I^\e_{n,k}(f)$,
$f\in\Cal F$, under the condition $(\e_1,\dots,\e_n)=u$ is the same
as the distribution of the random field we obtain by putting 
$u_l=\e_l$, $1\le l\le n$, in all coordinates $\e_l$ of the random 
variables $\tilde I^\e_{n,k}(f)$. On the other hand, the random 
variables we get in such a way agree with the random variables 
appearing in the sum defining $\tilde I_{n,k}(f|u)$, only the terms 
in this sum are listed in a different order. Lemma~15.1 is proved.

\medskip
Next we prove the following generalization of Lemma~7.1.

\medskip\noindent
{\bf Lemma 15.2. (Generalized version of the Symmetrization Lemma).}
{\it Let $Z_p$ and $\bar Z_p$, $p=1,2,\dots$, 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 $Z_p$, $p=1,2,\dots$, are $\Cal B$ measurable, and the 
inequality
$$
P(|\bar Z_p|\le\alpha|\Cal B)(\oo)\ge\beta\quad \text{for all }
\,p=1,2,\dots \text{ if }\,\oo\in B \tag15.3
$$
holds.
Then
$$
P\(\sup_{1\le p<\infty}|Z_p|>\alpha+u\)\le\frac1\beta P\(\supp_{1\le
p<\infty}|Z_p-\bar Z_p|>u\)+(1-P(B))\quad\text{for all } u>0.
\tag15.4
$$
}

\medskip\noindent
{\it Proof of Lemma 15.2.}\/ Put $\tau=\min\{p\colon\; |Z_p|>\alpha+u)$
if there exists such an index $p\ge1$, and put $\tau=0$ otherwise. Then
$$
\align
P(\{\tau=p\}\cap B)&\le\int_{\{\tau=p\}\cap B}\frac1\beta
P(|\bar Z_p|\le \alpha|\Cal B)\,dP
=\frac1\beta P(\{\tau=p\}\cap\{|\bar Z_p|\le\alpha\}\cap B)\\
&\le \frac1\beta P(\{\tau=p\}\cap\{|Z_p-\bar Z_p|>u\})
\quad \text{for all } p=1,2,\dots.
\endalign
$$
Hence
$$
\align
&P\(\sup_{1\le p<\infty}|Z_p|>\alpha+u\)-(1-P(B))\le
P\(\left\{\sup_{1\le p<\infty}|Z_p|>\alpha+u\right\}\cap B\) \\
&\qquad=\sum_{p=1}^\infty P(\{\tau=p\}\cap B)
\le \frac1\beta \sum_{p=1}^\infty P(\{\tau=p\}\cap\{|Z_p-\bar
Z_p|>u\}) \\
&\qquad \le\frac1\beta P\(\supp_{1\le p<\infty}|Z_p-\bar Z_p|>u\).
\endalign
$$
Lemma~15.2 is proved.

\medskip
To find a symmetrization argument useful in the proof of 
Proposition~$14.2'$ we want to bound the probability 
$P\(\supp_{f\in\Cal F}|\bar I_{n,k}(f)|>u\)$ by  
$C\cdot P\(\supp_{f\in\Cal F}|\tilde I_{n,k}(f)|>c u\)$ plus
a negligible error term with some appropriate numbers 
$C<\infty$ and $0<c<1$. The random variables $\bar I_{n,k}(f)$ 
and $\tilde I_{n,k}(f)$ appearing in these formulas were 
defined in~(14.11) and~(15.1). (Actually we work with a 
slightly modified version of formula~(14.11) where the random 
variables $\xi_l^{(j)}$ are replaced by the random variables 
$\xi_l^{(j,1)}$.) We shall prove the above mentioned estimate
with the help of Lemma~15.2. To find the random variables~$Z_p$ 
and~$\bar Z_p$ we want to work with in Lemma~15.2 let us list 
the elements of the class of functions $\Cal F$ as 
$\Cal F=\{f_1,f_2,\dots\}$. We shall apply Lemma~15.2 with the 
choice $Z_p=\bar I_{n,k}(f_p)$ and 
$\bar Z_p=\bar I_{n,k}(f_p)-\tilde I_{n,k}(f_p)$, $p=1,2,\dots$, 
together with the $\sigma$-algebra
$\Cal B=\Cal B(\xi_l^{(j,1)},\,1\le l\le n,\,1\le j\le k\}$.

Let us observe that $Z_p$ is a decoupled $U$-statistic depending 
on the random variables $\xi_l^{(j,1)}$, $1\le j\le k$, 
$1\le l\le n$, while $\bar Z_p$ is a linear combination of 
decoupled $U$-statistics, whose arguments contain not only the 
random variables of the form $\xi_l^{(j,-1)}$, but also the 
random variables of the form $\xi_l^{(j,1)}$. As a consequence, 
the random variables~$Z_p$ and~$\bar Z_p$ are not independent. 
This is the reason why we cannot apply Lemma~7.2 in the proof of
Proposition~$14.2'$.

We shall show that Lemma~15.2 with the choice of the above defined 
random variables $Z_p$ and $\bar Z_p$ and the $\sigma$-algebra 
$\Cal B$ may help us to prove the estimates we need in our
considerations. To apply this lemma we have to show that 
condition~(15.3) holds with an appropriate pair of numbers 
$(\alpha,\beta)$ and a $\Cal B$ measurable set~$B$ of 
probability almost~1. To check this condition is a hard but 
solvable problem. 

In Lemma~7.2 condition~(7.1) played a role similar to the
condition~(15.3) in Lemma~15.2. In that case we could check this 
condition by estimating the second moment $E\bar Z_n^2$. In the 
present case we shall estimate the supremum 
$\supp_{f_p\in\Cal F}E(\bar Z_p^2|\Cal B)$ of conditional second 
moments. In this formula $\bar Z_p$ is a (complicated) random 
variable depending on the function $f_p\in\Cal F$. The estimation 
of the supremum of the conditional second moments we want to work 
with is a hard problem, and the main difficulties of our proof 
appear at this point.

The conditional second moments whose supremum we want to estimate 
can be expressed as the integral of a random function that 
can be written down explicitly. In such a way we get a problem
similar to our original one about the estimation of 
$\supp_{f\in\Cal F}\bar I_{n,k}(f)$. It turned out that these 
two problems can be handled similarly. We can work out a 
symmetrization argument with the help of Lemma~15.2 in both 
cases, and an inductive argument similar to Proposition~7.3 can 
be formulated and proved which supplies the results we want 
to prove.

\medskip
We shall prove Proposition~$14.2'$ as a consequence of two inductive
propositions formulated in Propositions~15.3 and~15.4. Here we 
apply an approach similar to the proof of Proposition~6.2 with the
help of Proposition~7.3. The main difference is that now we have to
prove two inductive propositions simultaneously, because we have to
bound the supremum of some conditional variances, which demands
special attention. To formulate these propositions first we introduce
the notions of {\it good tail behaviour for a class of 
decoupled $U$-statistics}\/ and  {\it good tail behaviour for a 
class of integrals of decoupled $U$-statistics}.

\medskip\noindent
{\bf Definition of good tail behaviour for a class of decoupled
$U$-statistics.} {\it Let some measurable space $(X,\Cal X)$ be
given together with a probability measure $\mu$ on it. Let us
consider some countable 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~$n\ge k$ and a 
positive number $0<\sigma\le1$, and take $k$ independent copies 
$\xi_1^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$, 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 $\bar I_{n,k}(f)$, $f\in\Cal F$, defined 
in formula~(14.11). 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~$T$ (with 
parameters $n$ and $\sigma^2$ which are fixed in the sequel) if
$$
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>T.  \tag15.5
$$
}

\medskip\noindent
{\bf Definition of good tail behaviour for a class of integrals of
decoupled $U$-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~$n\ge k$ and a positive number $0<\sigma\le1$, and consider 
some countable 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 $\xi_1^{(j)},\dots,\xi_n^{(j)}$, $1\le j\le k$, 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 $\bar I_{n,k}(f,y)=\bar I_{n,k}(f_y)$
by means of these random variables $\xi_1^{(j)},\dots,\xi_n^{(j)}$,
$1\le j\le k$, the kernel function
$f_y(x_1,\dots,x_k)=f(x_1,\dots,x_k,y)$ and formula~(14.11). Define
with the help of these $U$-statistics $\bar I_{n,k}(f,y)$ the random
integrals
$$
H_{n,k}(f)=\int \bar I_{n,k}(f,y)^2\rho(\,dy), \quad f\in\Cal F.
\tag15.6
$$
Choose some real number $T>0$. We say that the set of random
integrals $H_{n,k}(f)$, $f\in\Cal F$, has a good tail behaviour at
level $T$ (with parameters $n$ and $\sigma^2$ which we fix in the
sequel) if
$$
P\(\sup_{f\in\Cal F} n^{-k}H_{n,k}(f)\ge A^2 n^k\sigma^{2k+2}\)
\le \exp\left\{-A^{1/(2k+1)}n\sigma^2 \right\}
\quad \text{for all } A> T. \tag15.7
$$
}

\medskip
Propositions~15.3 and~15.4 will be formulated with the help of the
above notions.

\medskip\noindent
{\bf Proposition 15.3.} {\it Let us fix a positive 
integer~$n\ge\max(k,2)$, a real number $0<\sigma\le2^{-(k+1)}$, a 
probability measure $\mu$ on a measurable space $(X,\Cal X)$ 
together with two real numbers $L\ge1$ and $D\ge1$ such that 
$n\sigma^2\ge L\log n+\log D$. Let us consider those 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 exponent~$L$
and parameter~$D$ for which all functions $f\in\Cal F$ satisfy the 
inequalities $\supp_{x_j\in X, 1\le j\le k}
|f(x_1,\dots,x_k)|\le 2^{-(k+1)}$ and $\int
f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)\le\sigma^2$.

There is some real number $A_0=A_0(k)>1$  such that if for all 
classes of functions $\Cal F$ which satisfy the  above conditions 
the sets of decoupled $U$-statistics $\bar I_{n,k}(f)$, $
f\in\Cal F$, have a good tail behaviour at  level~$T^{4/3}$ for 
some $T\ge A_0$, then they also have a good tail behaviour at 
level~$T$.}

\medskip\noindent
{\bf Proposition 15.4.} {\it Fix some positive integer 
$n\ge\max(k,2)$, a real number $0<\sigma\le2^{-(k+1)}$, 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 together with
two real numbers $L\ge1$ and $D\ge1$ such that 
$n\sigma^2>L\log n+\log D$ hold.

Let us consider those countable $L_2$-dense classes $\Cal F$ 
consisting of canonical functions $f(x_1,\dots,x_k,y)$ on the 
product  space $(X^k\times Y,\Cal X^k\times\Cal Y)$ with 
exponent $L\ge1$ and parameter $D\ge1$ whose elements 
$f\in\Cal F$ satisfy the inequalities
$$
\supp_{x_j\in X, 1\le j\le k, y\in Y}|f(x_1,\dots,x_k,y)|\le
2^{-(k+1)} \tag15.8
$$
and
$$
\int f^2(x_1,\dots,x_k,y)\mu(\,dx_1)\dots\mu(\,dx_k)\rho(\,dy)
\le\sigma^2 \quad  \text{for all } f\in \Cal F. \tag15.9
$$

There exists some number $A_0=A_0(k)>1$ such that if for all
classes of functions $\Cal F$ which satisfy the above conditions the
random integrals $H_{n,k}(f)$, $f\in\Cal F$, defined in (15.6) have a
good tail behaviour at level $T^{(2k+1)/2k}$ with some $T\ge A_0$,
then they also have a good tail behaviour at level~$T$.}

\medskip\noindent
{\it Remark:}\/ To complete the formulation of Proposition~15.4 we 
still have to clarify when we call a function $f(x_1,\dots,x_k,y)$
defined on the product space 
$(X^k\times Y,\Cal X^k\times\Cal Y,\mu^k\times\rho)$ canonical. 
Here the definition is slightly differs from that given
in formula~(8.8). 

We say that a function
$f(x_1,\dots,x_k,y)$ on the product space $(X^k\times Y,\Cal
X^k\times\Cal Y,\mu^k\times\rho)$ is canonical if
$$
\align
&\int f(x_1,\dots,x_{j-1},u,x_{j+1},\dots,x_k,y)\mu(\,du)=0\\
&\qquad \qquad \text{for all } 1\le j\le k,\; x_s\in X,
\;s\neq j \text{ and }y\in Y.
\endalign
$$
We do not require the analogous identity if we integrate with
respect to the variable $Y$ wit fixed arguments
$x_j\in X$,   $1\le j\le k$.

\medskip
Let me also remark that the estimate (15.7) we have formulated in 
the definition of the property `good tail behaviour for a class of
integrals of $U$-statistics' is fairly natural. We have applied the
natural normalization, and with such a normalization it is natural to
expect that the tail behaviour of the distribution of 
$\supp_{f\in\Cal F}n^{-k}H_{n,k}(f)$ is similar to that of 
$\const\(\sigma\eta^k\)^2$, where $\eta$ is a standard normal random 
variable. Formula~(15.7) expresses such a behaviour, only the power 
of the number~$A$ in the exponent at the right-hand side was chosen 
in a non-optimal way. Formula~(15.5) in the formulation of the 
property `good tail behaviour for a class of decoupled $U$-statistics' 
has a similar interpretation. It says that
$\supp_{f\in\Cal F}|n^{-k/2}I_{n,k}(f)|$ behaves similarly to
$\const\sigma|\eta^k|$ with a standard normal random variable $\eta$.

We wanted to prove the property of good tail behaviour for a class
of integrals of decoupled $U$-statistics under appropriate, not too
restrictive conditions. Let me remark that in Proposition~15.4  we
have imposed beside formula (15.8) a fairly weak condition (15.9)
about the $L_2$-norm of the function~$f$. Most difficulties appear
in the proof, because we did not want to impose more restrictive
conditions.

It is not difficult to derive Proposition~$14.2'$ from
Proposition~15.3. Indeed, let us observe that the set of decoupled
$U$-statistics determined by a class of functions $\Cal F$
satisfying the conditions of Proposition~15.3 has a good
tail-behaviour at level $T_0=\sigma^{-(k+1)}$, since under the
conditions of this Proposition the probability at the left-hand
side of (15.5) equals zero for $A>\sigma^{-(k+1)}$. Then we get
from Proposition~15.3 by induction with respect to the number $j$,
that this set of decoupled $U$-statistics has a good tail-behaviour
also for all $T=T_j=\ge T_0^{(3/4)^j}=\sigma^{-(k+1)(3/4)^j}$,
$j=0,1,2,\dots$, with such indices~$j$ for which 
$T_j=\sigma^{-(k+1)(3/4)^j}\ge A_0$. This implies that if a class of 
functions $\Cal F$ satisfies the conditions of Proposition~15.3, 
then the set of decoupled $U$-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~$14.2'$, 
only it has to be applied for the class of function 
$\Cal F'=\{2^{-(k+1)}f,\; f\in\Cal F\}$ instead of the original
class of functions $\Cal F$ which appears in Proposition~$14.2'$
with the same parameters~$\sigma$, $L$ and~$D$.

Similarly to the above argument an inductive procedure yields a
corollary of Proposition~15.4 formulated below. Actually, we shall
need this corollary of Proposition~15.4.

\medskip\noindent
{\bf Corollary of Proposition 15.4.} {\it If the class of functions
$\Cal F$ satisfies the conditions of Proposition~15.4, then there
exists a constant $\bar A_0=\bar A_0(k)>0$ depending only on $k$
such that the class of integrals $H_{n,k}(f)$, $f\in \Cal F$, defined
in formula (15.6) have a good tail behaviour at level $\bar A_0$.}

\medskip
The main difficulty in the proof of Proposition 15.3 arises in the
application of the symmetrization procedure corresponding to
Lemma~7.2 in the one-variate case. This difficulty can be overcome
by means of Proposition~15.4, more precisely by means of its
corollary. It helps us to estimate the conditional variances of
the decoupled $U$-statistics we have to handle in the proof of
Proposition~15.3. The proof of Propositions~15.3 and~15.4 apply
similar arguments, and they will be proved simultaneously. The 
following inductive procedure will be applied in their proof.
First Proposition~15.3 and then Proposition~15.4 is proved for 
$k=1$. If Propositions~15.3 and~15.4 are already proved
for all $k'<k$ for some number~$k$, then first we prove
Proposition~15.3 and then Proposition~15.4 for this number~$k$.

The proof both of Proposition 15.3 and 15.4 applies a symmetrization
argument that will be proved in Section~16. In Section~17
Propositions~15.3 and~15.4 will be proved with its help. 
They imply Proposition~$14.2'$, hence also Theorem~8.4.

%The expression $\tilde I^\e_{n,k}(f)$ defined in formula (15.2) is
%the linear combination (with coefficients $\pm1$) of expressions
%defined by a sum depending on a vector $v\in\Cal V_k$. These
%expressions, appearing in the inner sum of formula~(15.2), have the
%same distribution as the randomized decoupled $U$-statistic
%$\bar I^\e_{n,k}(f)$ defined in relation (14.12) if the random
%variables $\xi^{(j)}_l$ in formula (14.12) have the same
%distribution as the random variables
%$\xi^{(j,\pm1)}_l$, $1\le j\le k$, $1\le l\le n$. Hence a good
%estimate on the tail-distribution of the supremum of randomized
%decoupled $U$-statistic $\supp_{f\in\Cal F}\bar I^\e_{n,k}(f)$,
%$f\in\Cal F$, also provides an estimate on the tail distribution
%of $\supp_{f\in\Cal F}\tilde I^\e_{n,k}(f)$, or by Lemma~15.1
%of $\supp_{f\in\Cal F}\tilde I_{n,k}(f)$. We try to apply a
%symmetrization argument in the proof of Proposition~$14.2'$ which
%enables us to give a tail estimate of the distribution of 
%$\supp_{f\in\Cal F}\bar I_{n,k}(f)$ by means of a tail
%estimation of the supremum of the randomized decoupled 
%$U$-statistics $\supp_{f\in\Cal F}\bar I^\e_{n,k}(f)$. This can
%be done with the help of the previous
%arguments if we also can bound the tail-distribution of the
%supremum $\supp_{f\in\Cal F}\bar I_{n,k}(f)$ of the decoupled
%$U$-statistics introduced in (14.11) by means of the
%tail-distribution of the random variable 
%$\supp_{f\in\Cal F}\tilde I_{n,k}(f)$ defined in~(15.1).
