Let a sequence of iid. random variables ξ
1
, …, ξ
n
be given on a measurable space (
X,X
) with distribution µ together with a function
f
(
x1
, …,
xk
) on the product space (
Xk
,
Xk
). Let µ
n
denote the empirical measure defined by these random variables and consider the random integral \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$J_{n,k} (f) = \frac{{n^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{k!}}\smallint 'f(u_1 ,...,u_k )(\mu _n (du_1 ) - \mu (du_1 ))...(\mu _n (du_k ) - \mu (du_k )),$$
\end{document} where prime means that the diagonals are omitted from the domain of integration. A good bound is given on the probability
P
(|
Jn,k
(
f
)| >
x
) for all
x
> 0 which is similar to the estimate in the analogous problem we obtain by considering the Gaussian (multiple) Wiener-Itô integral of the function
f
. The proof is based on an adaptation of some methods of the theory of Wiener-Itô integrals. In particular, a sort of diagram formula is proved for the random integrals
Jn,k
(
f
) together with some of its important properties, a result which may be interesting in itself. The relation of this paper to some results about
U
-statistics is also discussed.
An estimate about multiple stochastic integrals with respect to a normalized empirical measure
