Central Limit Theorems for Empirical Measures

Let (X, A, P) be a probability space. Let X₁, X₂, , be independent X-valued random variables with distribution P. Let Pₙ: = n^-1 (ₗ䃑 + + ₗ䂸) be the empirical measure and let ₙ: = n¹2 (Pₙ - P). Given a class C a, we study the convergence in law of ₙ, as a stochastic process indexed by C, to a certain Gaussian process indexed by C. If convergence holds with respect to the supremum norm ₂ ₂|f (C) |, in a suitable (usually nonseparable) function space, we call C a Donsker class. For measurability, X may be a complete separable metric space, a = 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 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, ₘ (P (Cₘ) (1 - P (Cₘ) ) ) ʳ <.