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.
Suppose a1 , a2 ,... is a sequence of real numbers with an → ∞. If lim sup(X1+ ... + Xn)/an = 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+ ... + Yn)/an = 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.