#### Recommended Citation

Published in *Proceedings of the American Mathematical Society*, Volume 89, Issue 4, December 1, 1983, pages 685-690. 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 *a*_{1} , *a*_{2} ,... is a sequence of real numbers with *a*_{n} → ∞. If lim sup(*X*_{1}+ ... + *X _{n}*)/

*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}+ ... +

*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.

#### Disciplines

Mathematics

#### Included in

**URL:** https://digitalcommons.calpoly.edu/rgp_rsr/59