Almost Sure Stability of Partial Sums of Uniformly Bounded Random Variables

Theodore P. Hill, Georgia Institute of Technology - Main Campus

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Copyright © 1983 American Mathematical Society. The definitive version is available at http://www.jstor.org/stable/2044606.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

Suppose a₁ , a₂ ,... is a sequence of real numbers with a_n → ∞. If lim sup(X₁+ ... + X_n)/a_n = α 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 Y₁,Y₂, ... , lim sup(Y₁+ ... + Y_n)/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^-2_n < ∞ , then the assumption of nonnegativity may be dropped.