Published in The Annals of Probability, Volume 10, Issue 3, August 1, 1982, pages 828-830. Copyright © 1982 Institute of Mathematical Statistics. The definitive version is available at http://www.jstor.org/stable/2243392.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
Suppose that for every independent sequence of random variables satisfying some hypothesis condition H, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition H. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences.