Strong consistencies of the bootstrap moments

Abstract

Let X be a real valued random variable with E|X|r+δ0 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.

Keywords

• bootstrap sample size
• sample moment
• convergence with probability 1.