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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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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