UNIFORM DONSKER CLASSES OF FUNCTIONS

A class F of measurable functions on a probability space (A, A, P) is called a P-Donsker class and we also write F is-an-element-of CLT(P), if the empirical processes X(n)P = square-root n (P(n) - P) converge weakly to a P-Brownian bridge G(P) having bounded uniformly continuous sample paths almost surely. If this convergence holds for every probability measure P on (A, A), then F is called a universal Donsker class and we write F is-an-element-of CLT(M), where M = {all probability measures on (A, A)}. If the convergence holds uniformly in all P, then F is called a uniform Donsker class and we write F is-an-element-of CLT(u)(M). For many applications the latter concept is too restrictive and it is useful to focus instead on a fixed subcollection P of the collection M of all probability measures on (A, A). If the empirical processes converge weakly to G(P) uniformly for all P is-an-element-of P, then we say that F is a P-uniform Donsker class and write F is-an-element-of CLT(u)(P). We give general sufficient conditions for the P-uniform Donsker property and establish basic equivalences in the uniform (in P is-an-element-of P) central limit theorem for X(n), including a detailed study of the equivalences to the "functional" or "process in n" formulations of the CLT. We give applications of our uniform convergence results to sequences of measures {P(n)} and to bootstrap resampling methods.