The Annals of Probability

Asymptotic Properties of the Bootstrap for Heavy-Tailed Distributions

Peter Hall

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Abstract

We establish necessary and sufficient conditions for convergence of the distribution function of a bootstrapped mean, suitably normalized. It turns out that for convergence to occur, the sampling distribution must either be in the domain of attraction of the normal distribution or have slowly varying tails. In the first case the limit is normal; in the latter, Poisson. Between these two extremes of light tails and extremely heavy tails, the bootstrap distribution function of the mean does not converge in probability to a nondegenerate limit. However, it may converge in distribution. We show that when there is no convergence in probability, a small number of extreme sample values determine behaviour of the bootstrap distribution function. This result is developed and used to interpret recent work of Athreya.

Article information

Source
Ann. Probab., Volume 18, Number 3 (1990), 1342-1360.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990748

Digital Object Identifier
doi:10.1214/aop/1176990748

Mathematical Reviews number (MathSciNet)
MR1062071

Zentralblatt MATH identifier
0714.62035

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks 62G05: Estimation

Keywords
Bootstrap central limit theorem domain of attraction heavy tail normal distribution stable law

Citation

Hall, Peter. Asymptotic Properties of the Bootstrap for Heavy-Tailed Distributions. Ann. Probab. 18 (1990), no. 3, 1342--1360. doi:10.1214/aop/1176990748. https://projecteuclid.org/euclid.aop/1176990748


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