The strong law of large numbers for sums of randomly chosen random variables

Abstract

Let {Xn,n ≥ 1} be a sequence of independent or identically distributed dependent random variables, and let {An,n ≥ 1} be a sequence of random subsets of natural numbers independent of {Xn, n ≥ 1}. In this paper, we describe the strong law of large numbers (SLLN) of the form ∑i∈An(Xi−E∑i∈AnXi)/bn→0a.s. as n → ∞ for some sequence of nondecreasing positive numbers {bn, n ≥ 1}. There often arises an assumption that {An, n ≥ 1} are almost surely increasing: An ⊂ An + 1, a. s n ≥ 1.