The Annals of Probability

Bootstrapping General Empirical Measures

Evarist Gine and Joel Zinn

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Abstract

It is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions $\mathscr{F}$ and based on a probability measure $P$ holds a.s. if and only if $\mathscr{F} \in \mathrm{CLT}(P)$ and $\int F^2 dP < \infty$, where $F = \sup_{f \in \mathscr{F}}|f|$, and it holds in probability if and only if $\mathscr{F} \in \mathrm{CLT}(P)$. Thus, for a large class of statistics, no local uniformity of the CLT (about $P$) is needed for the bootstrap to work. Consistency of the bootstrap (the bootstrapped law of large numbers) is also characterized. (These results are proved under certain weak measurability assumptions on $\mathscr{F}$.)

Article information

Source
Ann. Probab., Volume 18, Number 2 (1990), 851-869.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990862

Digital Object Identifier
doi:10.1214/aop/1176990862

Mathematical Reviews number (MathSciNet)
MR1055437

Zentralblatt MATH identifier
0706.62017

JSTOR
links.jstor.org

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 62E20: Asymptotic distribution theory 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Bootstrapping empirical processes central limit theorem law of large numbers

Citation

Gine, Evarist; Zinn, Joel. Bootstrapping General Empirical Measures. Ann. Probab. 18 (1990), no. 2, 851--869. doi:10.1214/aop/1176990862. https://projecteuclid.org/euclid.aop/1176990862


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