International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 797-802
doi:10.1155/S0161171291001060
Strong consistencies of the bootstrap moments
1Department of Mathematics, National Tsing Hua University, Hsinchu 3004, Taiwan
2Department of Statistics, University of North Carolina, Chapel Hill 27599, NC, USA
Received 14 November 1990; Revised 23 January 1991
Copyright © 1991 Tien-Chung Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let X be a real valued random variable with E|X|r+δ<∞ for some positive
integer r and real number, δ, 0<δ≤r, and let {X,X1,X2,…} be a sequence of
independent, identically distributed random variables. In this note, we prove that,
for almost all w∈Ω, μr;n*(w)→μr with probability 1. if limn→∞infm(n)n−β>0 for
some β>r−δr+δ, where μr;n* is the bootstrap rth sample moment of the bootstrap sample some
with sample size m(n) from the data set {X,X1,…,Xn} and μr is the rth moment of
X. The results obtained here not only improve on those of Athreya [3] but also the
proof is more elementary.