Article

Almost Sure Stability of Partial Sums of Uniformly Bounded Random Variables

Proceedings of the American Mathematical Society
Publication Date

12-1-1983
Abstract

Suppose *a*1 , *a*2 ,... is a sequence of real numbers with *a*n → ∞. If lim sup(*X*1+ ... + *Xn*)/*a*n = α a.s. for every sequence of independent nonnegative uniformly bounded random variables *X*1,*X*2,... satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables *Y*1,*Y*2, ... , lim sup(*Y*1+ ... + *Yn*)/*a*n = α a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, Σ∞*a*-2n < ∞ , then the assumption of nonnegativity may be dropped.

Disciplines

Citation Information

Theodore P. Hill. "Almost Sure Stability of Partial Sums of Uniformly Bounded Random Variables" *Proceedings of the American Mathematical Society*Vol. 89 Iss. 4 (1983) p. 685 - 690

Available at: http://works.bepress.com/tphill/57/