Rates of convergence in the central limit theorem for empirical processes

Abstract

In this paper we study we uniform behavior of the empirical brownian bridge over families of functions F bounded by a function F (the observations are independent with common distribution P). Under some suitable entropy conditions which were already used by Kolčinskii and Pollard, we prove exponential inequalities in the uniformly bounded case where F is a constant (the classical Kiefer's inequality (1961) is improved), as well as weak and strong invariance principles with rates of convergence in the case where F belongs to L 2+δ(P) with δε]0,1] (our results improve on Dudley, Philipp's results (1983) whenever F is a Vapnik-Červonenkis class in the uniformly bounded case and are new in the unbounded case).Key words and phrasesInvariance principlesempirical processesgaussian processesexponential bounds