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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Summability methods for independent identically distributed random variables


Author: Tze Leung Lai
Journal: Proc. Amer. Math. Soc. 45 (1974), 253-261
MSC: Primary 60F15
DOI: https://doi.org/10.1090/S0002-9939-1974-0356194-4
MathSciNet review: 0356194
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Abstract: In this paper, we present certain theorems concerning the Cesaro $ (C,\alpha )$, Abel $ (A)$, Euler $ (E,q)$ and Borel $ (B)$ summability of $ \Sigma {Y_i}$, where $ {Y_i} = {X_i} - {X_{i - 1}},{X_0} = 0$ and $ {X_1},{X_2}, \cdots $ are i.i.d. random variables. While the Kolmogorov strong law of large numbers and the Hartman-Wintner law of the iterated logarithm are related to $ (C,1)$ summability and involve the finiteness of, respectively, the first and second moments of $ {X_1}$, their analogues for Euler and Borel summability involve different moment conditions, and the analogues for $ (C,\alpha )$ and Abel summability remain essentially the same.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0356194-4
Keywords: Summability, strong law of large numbers, law of the iterated logarithm
Article copyright: © Copyright 1974 American Mathematical Society