Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely

Abstract

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