An estimate on the maximum of a nice class of stochastic integrals

Abstract

Let a sequence of iid. random variables $\xi_1,...,\xi_n$ be given on a space $(X,\cal X)$ with distribution $\mu$ together with a nice class $\cal F$ of functions $f(x_1,...,x_k)$ of $k$ variables on the product space $(X^k,{\cal X}^k)$. For all $f\in\cal F$ we consider the random integral $J_{n,k}(f)$ of the function $f$ with respect to the $k$-fold product of the normalized signed measure $\sqrt n(\mu_n-\mu)$, where $\mu_n$ denotes the empirical measure defined by the random variables $\xi_1,...,\xi_n$ and investigate the probabilities $P(\sup_{f\in {\cal F}}|J_{n,k}(f)|>x)$ for all $x>0$. We show that for nice classes of functions, for instance if $\cal F$ is a Vapnik-Cervonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered.