An estimate on the supremum of a nice class of stochastic integrals and U-statistics

Abstract

Let a sequence of iid. random variables ξ 1, . . . ,ξ n be given on a space [InlineMediaObject not available: see fulltext.] with distribution μ together with a nice class [InlineMediaObject not available: see fulltext.] of functions f(x 1, . . . ,x k ) of k variables on the product space [InlineMediaObject not available: see fulltext.] For all f [InlineMediaObject not available: see fulltext.] 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 [InlineMediaObject not available: see fulltext.] where μ n denotes the empirical measure defined by the random variables ξ 1, . . . ,ξ n and investigate the probabilities [InlineMediaObject not available: see fulltext.] for all x>0. We show that for nice classes of functions, for instance if [InlineMediaObject not available: see fulltext.] is a Vapnik-Červonenkis 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. A similar result holds for degenerate U-statistics, too. © 2005 Springer-Verlag Berlin Heidelberg.