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 \( {\sum}_{i\in {A}_n}\left({X}_i-\mathrm{E}{\sum}_{i\in {A}_n}{X}_i\right)/{b}_n\to 0\ \mathrm{a}.\mathrm{s}. \) 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.
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28 October 2021
A Correction to this paper has been published: https://doi.org/10.1007/s10986-021-09544-7
21 November 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10986-022-09582-9
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Gdula, A.M., Krajka, A. The strong law of large numbers for sums of randomly chosen random variables. Lith Math J 61, 471–482 (2021). https://doi.org/10.1007/s10986-021-09528-7
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DOI: https://doi.org/10.1007/s10986-021-09528-7