Almost sure stability of partial sums of uniformly bounded random variables

Abstract:

Suppose ${a_1}$, ${a_2}, \ldots$ is a sequence of real numbers with ${a_n} \to \infty$. If $({X_1} + \cdots + {X_n})/{a_n} = \alpha$ a.s. for every sequence of independent nonnegative uniformly bounded random variables ${X_1}$, ${X_2}, \ldots$ satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables ${Y_1}$, ${Y_2}, \ldots ,$, $\lim {\text { sup}}({Y_1} + \cdots + {Y_n})/{a_n} = \alpha$ a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, $\Sigma ^\infty a_n^{ - 2} < \infty$, then the assumption of nonnegativity may be dropped.