Strong law for the bootstrap

Abstract

Let X₁, X₂, X₃, ... be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,...) and integers n and m, construct Y_n,i, I = 1, 2, ..., m as i.i.d. r.v. with conditional distribution P∗(Y_n,i = X_j) = 1/n for 1≤ j ≤ n. (P∗ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of (1/m)∑^m_i=1 Y_n,i to μ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).