Strong consistencies of the bootstrap moments - Tien-Chung Hu (International Journal of Mathematics and Mathematical Sciences)
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.