An estimate about multiple stochastic integrals with respect to a normalized empirical measure

Abstract

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.