International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 797-802

Strong consistencies of the bootstrap moments

Tien-Chung Hu1,2

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.


Let X be a real valued random variable with E|X|r+δ< for some positive integer r and real number, δ, 0<δr, and let {X,X1,X2,} be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost all wΩ, μr;n*(w)μr with probability 1. if limninfm(n)nβ>0 for some β>rδr+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.