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.