2003/10/20 by Péter Major, Peter Major, Major, Peter · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Mathematical Approximation and Integration #math.PR
paper · pdf · doi:10.48550/arxiv.math/0310324
This article can also be found at my homepage http://www.renyi.hu/~major/public1.html
arxiv created 2003/10/20 · arxiv updated 2009/12/01
Let a sequence of iid. random variables ξ1,...,ξn be given on a space (X,\cal X) with distribution μ together with a nice class \cal F of functions f(x1,...,xk) of k variables on the product space (Xk,\cal Xk). For all f∈\cal F we consider the random integral Jn,k(f) of the function f with respect to the k-fold product of the normalized signed measure √ n(μn-μ), where μn denotes the empirical measure defined by the random variables ξ1,...,ξn and investigate the probabilities P(sup_f∈ \cal F|Jn,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.