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(δX₁ + ⋯ + δXₙ) be the empirical measure and let νₙ := n¹/2(Pₙ - P). Given a class C ⊂ a, we study the convergence in law of νₙ, as a stochastic process indexed by C, to a certain Gaussian process indexed by C. If convergence holds with respect to the supremum norm C ∈ C|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 ₘ\ independent for P, then C is a Donsker class if and only if for some r, σₘ(P(Cₘ)(1 - P(Cₘ)))ʳ < ∞.