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Proceedings of the American Mathematical Society

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

 

 

Almost sure stability of partial sums of uniformly bounded random variables


Author: Theodore P. Hill
Journal: Proc. Amer. Math. Soc. 89 (1983), 685-690
MSC: Primary 60F15; Secondary 60G42
DOI: https://doi.org/10.1090/S0002-9939-1983-0718997-9
MathSciNet review: 718997
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Abstract: Suppose $ {a_1}$, $ {a_2}, \ldots $ is a sequence of real numbers with $ {a_n} \to \infty $. If $ ({X_1} + \cdots + {X_n})/{a_n} = \alpha $ a.s. for every sequence of independent nonnegative uniformly bounded random variables $ {X_1}$, $ {X_2}, \ldots $ satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables $ {Y_1}$, $ {Y_2}, \ldots ,$, $ \lim {\text{ sup}}({Y_1} + \cdots + {Y_n})/{a_n} = \alpha $ a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, $ \Sigma ^\infty a_n^{ - 2} < \infty $, then the assumption of nonnegativity may be dropped.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0718997-9
Keywords: Almost-sure stability of partial sums, martingale strong law of large numbers, conditional generalizations of strong laws
Article copyright: © Copyright 1983 American Mathematical Society