Open Access
April, 1993 Almost Sure Bootstrap of the Mean Under Random Normalization
Steven J. Sepanski
Ann. Probab. 21(2): 917-925 (April, 1993). DOI: 10.1214/aop/1176989274


We consider the problem of when the bootstrap sample mean, appropriately normalized and centered, converges in distribution along almost every sample path. We allow the normalizing sequence to be an arbitrary sequence of positive random variables. It is proved that the only possible normalizing sequence is essentially $(\sum^n_{i = 1}X^2_i)^{1/2}$. Furthermore, if the bootstrap sample mean converges along almost every sample path, then either the variance is finite or else the distribution of $X$ is extremely heavy tailed. In the latter case, the distribution of the bootstrap sample mean is completely determined by how many times the maximum order statistic from the original random sample is repeated in the bootstrap sample. The necessary condition on how heavy the tails must be is $(\sum^n_{i = 1}|X_i|^p)^{1/p}/(\sum^n_{i = 1}X^2_i)^{1/2} \rightarrow 1$ almost surely for all $p \in (0, \infty\rbrack$. Furthermore, we show that in this case the limit of the bootstrap sample mean normalized by $(\sum^n_{i = 1}X^2_i)^{1/2}$ is Poisson with mean 1.


Download Citation

Steven J. Sepanski. "Almost Sure Bootstrap of the Mean Under Random Normalization." Ann. Probab. 21 (2) 917 - 925, April, 1993.


Published: April, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0774.62019
MathSciNet: MR1217572
Digital Object Identifier: 10.1214/aop/1176989274

Primary: 62E20
Secondary: 60F05 , 62F12

Keywords: bootstrap , central limit theorem , maximum order statistic , slowly varying tails

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.21 • No. 2 • April, 1993
Back to Top