# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Almost sure stability of partial sums of uniformly bounded random variablesHTML articles powered by AMS MathViewer

by Theodore P. Hill
Proc. Amer. Math. Soc. 89 (1983), 685-690 Request permission

## 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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