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

Let {X(n,)n >= 1} be a sequence of independent or identically distributed dependent random variables, and let {A(n,)n >= 1} be a sequence of random subsets of natural numbers independent of {X-n, n >= 1}. In this paper, we describe the strong law of large numbers (SLLN) of the form Sigma(i is an element of An)(X-i - E Sigma(i is an element of An) X-i)/b(n) -> 0 a.s. as n -> infinity for some sequence of nondecreasing positive numbers {b(n), n >= 1}. There often arises an assumption that {A(n), n >= 1} are almost surely increasing: A(n) subset of A(n + 1), a. s n >= 1.