## The Annals of Probability

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

### Bootstrapping General Empirical Measures

#### 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