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The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences

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Abstract

This paper establishes complete convergence for weighted sums and the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables \(\{X,X_n,n\ge 1\}\) with general normalizing constants under a moment condition that \(ER(X)<\infty \), where \(R(\cdot )\) is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Stat Probab Lett 92:45–52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijn conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944–1946, 1995) on the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrative examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

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Acknowledgements

The authors are grateful to the referee for constructive, perceptive and substantial comments and suggestions which enabled us to greatly improve the paper. In particular, the referee’s suggestions on the asymptotic inverse of the regularly varying function \(x^\alpha L(x)\) enabled us to obtain Theorems 3.1 and 4.1 which are considerably more general than those of the initial version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Hoa Lu University, Ninh Binh, Vietnam

    Vu T. N. Anh

  2. Department of Mathematics, Vinh University, Nghe An, Vietnam

    Nguyen T. T. Hien, Le V. Thanh & Vo T. H. Van

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  1. Vu T. N. Anh
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  3. Le V. Thanh
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Correspondence to Le V. Thanh.

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The paper was supported by NAFOSTED, Grant No. 101.03-2015.11.

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Anh, V.T.N., Hien, N.T.T., Thanh, L.V. et al. The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences. J Theor Probab 34, 331–348 (2021). https://doi.org/10.1007/s10959-019-00973-2

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