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

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∈AnXi−E∑i∈AnXi/bn→0a.s.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\sum}_{i\in {A}_n}\left({X}_i-\mathrm{E}{\sum}_{i\in {A}_n}{X}_i\right)/{b}_n\to 0\ \mathrm{a}.\mathrm{s}. $$\end{document} 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.