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.