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

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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 \(1/\rho \) for some \(\rho \in ]0,1]\). 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 \(\rho \ne 1\). We also prove the necessity part in Gut’s weak law of large numbers when the summands are pairwise negatively dependent.

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Acknowledgements

The author is grateful to an anonymous referee for his/her insightful suggestions and comments which improved the quality of the paper drastically. This research is a contribution to the Project PRFU C00L03UN130120210002, funded by the DGRSDT-MESRS-Algeria.

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  1. Laboratoire de Statistique et Modélisations Aléatoires, Faculty of Sciences, Abou Bekr Belkaid University, BP 119, Tlemcen, 13000, Algeria

    Fakhreddine Boukhari

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  1. Fakhreddine Boukhari
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Correspondence to Fakhreddine Boukhari.

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Boukhari, F. On a Weak Law of Large Numbers with Regularly Varying Normalizing Sequences. J Theor Probab 35, 2068–2079 (2022). https://doi.org/10.1007/s10959-021-01120-6

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  • DOI: https://doi.org/10.1007/s10959-021-01120-6

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