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 $(X,\mathcal{X})$ with distribution $\mu$ together with a nice class $\mathcal{F}$ of functions $f(x_1,...,x_k)$ of $k$ variables on the product space $(X^k,{\mathcal{X}}^k)$. For all $f\in \mathcal{F}$ 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 $P({\sup }_{f\in \mathcal{F}}|J_{n,k}(f)|>x)$ for all $x>0$. We show that for nice classes of functions, for instance if $\mathcal{F}$ 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.

This article can also be found at my homepage http://www.renyi.hu/~major/public1.html

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