The Annals of Probability

Uniform Donsker Classes of Functions

Anne Sheehy and Jon A. Wellner

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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.

Article information

Source
Ann. Probab., Volume 20, Number 4 (1992), 1983-2030.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989538

Digital Object Identifier
doi:10.1214/aop/1176989538

Mathematical Reviews number (MathSciNet)
MR1188051

Zentralblatt MATH identifier
0763.60012

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62G05: Estimation 62G30: Order statistics; empirical distribution functions 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F17: Functional limit theorems; invariance principles

Keywords
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

Citation

Sheehy, Anne; Wellner, Jon A. Uniform Donsker Classes of Functions. Ann. Probab. 20 (1992), no. 4, 1983--2030. doi:10.1214/aop/1176989538. https://projecteuclid.org/euclid.aop/1176989538


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