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

Fakhreddine Boukhari ()
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Fakhreddine Boukhari: Faculty of Sciences, Abou Bekr Belkaid University

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 2068-2079

Abstract: 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.

Keywords: Kolmogorov–Feller weak law; General limit theorems; Pairwise NQD random variables; Slow variation; 60F15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01120-6

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