Uniform Donsker Classes of Functions

Abstract

A class $\mathscr{F}$ of measurable functions on a probability space $(A, \mathbb{A}, P)$ is called a $P$-Donsker class and we also write $\mathscr{F} \in \operatorname{CLT}(P)$, if the empirical processes $\mathbb{X}^P_n \equiv \sqrt{n}(\mathbb{P}_n - P)$ 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 $(A, \mathbb{A})$, then $\mathscr{F}$ is called a universal Donsker class and we write $\mathscr{F} \in \operatorname{CLT}(\mathbf{M})$, where $\mathbf{M} \equiv \{$all probability measures on $(A, \mathbb{A})\}$. If the convergence holds uniformly in all $P$, then $\mathscr{F}$ is called a uniform Donsker class and we write $\mathscr{F} \in \operatorname{CLT}_u(\mathbf{M})$. For many applications the latter concept is too restrictive and it is useful to focus instead on a fixed subcollection $\mathscr{P}$ of the collection $\mathbf{M}$ of all probability measures on $(A, \mathbb{A})$. If the empirical processes converge weakly to $G_P$ uniformly for all $P \in \mathscr{P}$, then we say that $\mathscr{F}$ is a $\mathscr{P}$-uniform Donsker class and write $\mathscr{F} \in \operatorname{CLT}_u(\mathscr{P})$. We give general sufficient conditions for the $\mathscr{P}$-uniform Donsker property and establish basic equivalences in the uniform (in $P \in \mathscr{P}$) central limit theorem for $\mathbf{X}_n$, including a detailed study of the equivalences to the "functional" or "process in $n$" formulations of the $\operatorname{CLT}$. We give applications of our uniform convergence results to sequences of measures $\{P_n\}$ and to bootstrap resampling methods.