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Sharp sufficient conditions for mean convergence of the maximal partial sums of dependent random variables with general norming sequences

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Abstract

This paper provides sharp sufficient conditions for mean convergence of the maximal partial sums from triangular arrays of dependent random variables with general norming sequences. As an application, we use this result to give a positive answer to an open question in [Test 32(1):74–106, 2023] concerning mean convergence for the maximal partial sums under regularly varying moment conditions. The techniques developed in the present work also enable us to establish a result on mean convergence for sums of pairwise negatively dependent random variables, which gives an improvement of the main result of Sung [Appl Math Lett 26(1):18–24, 2013] and Ordóñez Cabrera and Volodin [J Math Anal Appl 305(2):644–658, 2005].

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Acknowledgements

The author is grateful to two anonymous Reviewers for carefully reading the manuscript and for offering useful comments and suggestions which enabled him to improve the paper. Regarding Theorem 4.1, one of the Reviewers kindly raised a question as to whether or not the assumptions that \(L(\cdot )\) is increasing and \(L(x)\ge 1\) can be removed for the case \(1<p<2\). This inquiry led to Example 4.2.

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

    Lê Vǎn Thành

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Thành, L.V. Sharp sufficient conditions for mean convergence of the maximal partial sums of dependent random variables with general norming sequences. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 40 (2024). https://doi.org/10.1007/s13398-023-01540-5

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