Gaussian Characterization of Uniform Donsker Classes of Functions

Abstract

It is proved that, for classes of functions F satisfying some measurability, the empirical processes indexed by F and based on P ∈ P(S) satisfy the central limit theorem uniformly in P ∈ P(S) if and only if the P-Brownian bridges Gp indexed by F are sample bounded and ρp uniformly continuous uniformly in P ∈ P(S). Uniform exponential bounds for empirical processes indexed by universal bounded Donsker and uniform Donsker classes of functions are also obtained.