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Almost Sure Stability of Partial Sums of Uniformly Bounded Random Variables

Theodore P. Hill
Proceedings of the American Mathematical Society
Vol. 89, No. 4 (Dec., 1983), pp. 685-690
DOI: 10.2307/2044606
Stable URL: http://www.jstor.org/stable/2044606
Page Count: 6
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Almost Sure Stability of Partial Sums of Uniformly Bounded Random Variables
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Abstract

Suppose a1, a2,... is a sequence of real numbers with an → ∞. If $\lim \sup(X_1 + \cdots + X_n)/a_n = \alpha$ a.s. for every sequence of independent nonnegative uniformly bounded random variables X1, X2,... satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables $Y_1, Y_2,\ldots, \lim \sup(Y_1 + \cdots + Y_n)/a_n = \alpha$ a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, $\Sigma^\infty a^{-2}_n < \infty$, then the assumption of nonnegativity may be dropped.

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