On the law of large numbers for the bootstrap mean
type: Item
instance of: scholarly article
title: On the law of large numbers for the bootstrap mean
zbMATH Open document ID: 0752.62034
DOI: 10.1016/0167-7152(92)90203-H
published in: Statistics \& Probability Letters
publication date: 1993-01-16T00:00:00Z
review text: Let \(X_ 1,X_ 2,\dots\) be an infinite sequence of independent and identically distributed random variables. For each \(n=1,2,\dots\), let \(X^{(n)}_ 1,\dots,X^{(n)}_ m\) be the ordinary Efron bootstrap sample from \(X_ 1,\dots,X_ n\) with bootstrap sample size \(m=m(n)\). The author proves, among others, that \[ \lim_{n\to\infty} m^{- 1}\sum^ m_{j=1} X^{(n)}_ j=E(X)\qquad\text{almost surely} \] if for some \(\delta\) \((0\leq\delta<1)\) \[ E(| X|^{1+\delta})<\infty\qquad\text{and}\qquad\lim_{n\to\infty} n^{1-\delta}/m<\infty, \] or if for some \(r=1,2,\dots\) and \(\delta>0\), \[ E(X^ 2)<\infty\qquad\text{and}\qquad\limsup(\log n \log_ 2\cdots\log_{r-1}n (\log_ r n)^{1+\delta})/m(n)<\infty, \] where \(\log_ r\) stands for the \(r\)-times iterated logarithm. The results improve on those of \textit{K. Athreya} [ibid. 1, 147-150 (1983; Zbl 0518.62016)].
reviewed by: Ken-ichi Yoshihara
Mathematics Subject Classification ID: 62G09
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 93336
zbMATH Keywords: bootstrap sample means
zbMATH Keywords: minimal moment conditions
zbMATH Keywords: rate of divergence of the bootstrap sample size
zbMATH Keywords: law of large numbers
zbMATH Keywords: Efron bootstrap a sample
zbMATH Keywords: iterated logarithm
zbMATH Keywords: Efron bootstrap sample
author: Sándor Csörgő
MaRDI profile type: Publication
cites work: The bootstrap of the mean with arbitrary bootstrap sample size
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cites work: Q3794956
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cites work: Some Results on the Complete and Almost Sure Convergence of Linear Combinations of Independent Random Variables and Martingale Differences
full work available at URL: 0167-7152(92)90203-h
OpenAlex ID: W2113200496
description: scientific article; zbMATH DE number 93336