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On a Weak Law of Large Numbers with Regularly Varying Normalizing Sequences

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  • Fakhreddine Boukhari

    (Faculty of Sciences, Abou Bekr Belkaid University)

Abstract

The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables has been extended by Gut (J. Theoret. Probab. 17, 769–779, 2004) to the case where the normalizing sequence is regularly varying with index $$1/\rho $$ 1 / ρ for some $$\rho \in ]0,1]$$ ρ ∈ ] 0 , 1 ] . In this paper, we show that the sufficiency part in Gut’s theorem is valid without any restriction on the dependence structure of the underlying sequence, provided that $$\rho \ne 1$$ ρ ≠ 1 . We also prove the necessity part in Gut’s weak law of large numbers when the summands are pairwise negatively dependent.

Suggested Citation

  • Fakhreddine Boukhari, 2022. "On a Weak Law of Large Numbers with Regularly Varying Normalizing Sequences," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2068-2079, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01120-6
    DOI: 10.1007/s10959-021-01120-6
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    References listed on IDEAS

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    1. H. Naderi & P. Matuła & M. Amini & H. Ahmadzade, 2019. "A version of the Kolmogrov–Feller weak law of large numbers for maximal weighted sums of random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(21), pages 5414-5418, November.
    2. Fakhreddine Boukhari, 2021. "Weak laws of large numbers for maximal weighted sums of random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(1), pages 105-115, January.
    3. Rosalsky, Andrew, 1993. "On the almost certain limiting behavior of normed sums of identically distributed positive random variables," Statistics & Probability Letters, Elsevier, vol. 16(1), pages 65-70, January.
    4. Maller, R. A., 1980. "On the law of large numbers for stationary sequences," Stochastic Processes and their Applications, Elsevier, vol. 10(1), pages 65-73, June.
    5. A. Gut, 2004. "An Extension of the Kolmogorov–Feller Weak Law of Large Numbers with an Application to the St. Petersburg Game," Journal of Theoretical Probability, Springer, vol. 17(3), pages 769-779, July.
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