Strong consistencies of the bootstrap moments
Let be a real valued random variable with for some positive integer and real number, , , and let be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost all , with probability . if for some , where is the bootstrap sample moment of the bootstrap sample some with sample size from the data set and is the moment of . The results obtained here not only improve on those of Athreya  but also the proof is more elementary.
Copyright © 1991 Hindawi Publishing Corporation. 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.