Universal Donsker Classes and Metric Entropy

Abstract

Let $(X, \mathscr{A})$ be a measurable space and $\mathscr{F}$ a class of measurable functions on $X. \mathscr{F}$ is called a universal Donsker class if for every probability measure $P$ on $\mathscr{A}$, the centered and normalized empirical measures $n^{1/2}(P_n - P)$ converge in law, with respect to uniform convergence over $\mathscr{F}$, to the limiting "Brownian bridge" process $G_P$. Then up to additive constants, $\mathscr{F}$ must be uniformly bounded. Several nonequivalent conditions are shown to imply the universal Donsker property. Some are connected with the Vapnik-Cervonenkis combinatorial condition on classes of sets, others with metric entropy. The implications between the various conditions are considered. Bounds are given for the metric entropy of convex hulls in Hilbert space.