Rates of convergence in the central limit theorem for empirical processes

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 Kolcinskii 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 L2+δ(P) with δe]0,1] (our results improve on Dudley, Philipp's results (1983) whenever F is a Vapnik-Cervonenkis class in the uniformly bounded case and are new in the unbounded case).

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