Uniform Donsker Classes of Functions

Sheehy, Anne ; Wellner, Jon A.

Ann. Probab., Tome 20 (1992) no. 4, p. 1983-2030 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1992-10-14
Classification:  Bootstrap,  central limit theorem,  empirical process,  functional central limit theorem,  Gaussian processes,  parametric bootstrap,  nonparametric bootstrap,  regular estimators,  sequential empirical process,  uniformity,  uniform integrability,  weak approximation,  60F05,  62G05,  62G30,  60B12,  60F17