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Rates of convergence in the strong law of large numbers for weighted averages of nonidentically distributed random variables

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Abstract

Integral tests are found for the convergence of two Spitzer-type series associated with a class of weighted averages introduced by Jajte [On the strong law of large numbers, Ann. Probab., 31(1):409–412, 2003]. Our main theorems are valid for a large family of dependent random variables that are not necessarily identically distributed. As a byproduct, we improve the Marcinkiewicz–Zygmund strong law of large numbers for asymptotically almost negatively associated sequences due to Chandra and Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables Acta Math. Hung., 71(4):327–336, 1996]. We also complement two limit theorems recently derived by Anh et al. [TheMarcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, J. Theor. Probab., 34(1):331–348, 2021] and Thành [On a new concept of stochastic domination and the laws of large numbers, Test, 32(1):74–106, 2023]. The obtained results are new even when the summands are independent.

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  1. Laboratoire de Statistique et Modélisations Aléatoires, Faculté des Sciences, Université Abou Bekr Belkaid, BP 119, Tlemcen, 13000, Algérie

    Wahiba Benyahia & Fakhreddine Boukhari

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  1. Wahiba Benyahia
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  2. Fakhreddine Boukhari
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Correspondence to Wahiba Benyahia.

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Benyahia, W., Boukhari, F. Rates of convergence in the strong law of large numbers for weighted averages of nonidentically distributed random variables. Lith Math J 64, 1–17 (2024). https://doi.org/10.1007/s10986-024-09621-7

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