An estimate on the maximum of a nice class of stochastic integrals
Abstract
Description
Let a sequence of iid. random variables $ξ_1,...,ξ_n$ be given on a space $(X,\cal X)$ with distribution $μ$ 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 $\sqrt n(μ_n-μ)$, where $μ_n$ denotes the empirical measure defined by the random variables $ξ_1,...,ξ_n$ 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.
This article can also be found at my homepage http://www.renyi.hu/~major/public1.html
This article can also be found at my homepage http://www.renyi.hu/~major/public1.html