Central Limit Theorems for Empirical Measures

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Probability. Let (X, i, P) be a probability space. Let X,, *2 * *., be independent X-valued random variables with distribution P. Let P-n'-1(Ox, + * + &x) be the empirical measure and let Pn n n(Pn-P). Given a class (c 3, we study the convergence in law of ",,, as a stochastic process indexed by (P, to a certain Gaussian process indexed by (C. If convergence holds with respect to the supremum norm supc =-df(C)I, in a suitable (usually nonseparable) function space, we call (C a Donsker class. For measurability, X may be a complete separable metric space, &, = Borel sets, and (C a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some m, and every set F c X with m elements, (C does not cut all subsets of F (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If C(is a sequence { Cm) independent for P. then (C is a Donsker class if and only if for some r, Xm(P(Cm)(l-P(Cm)))r < oo.