On the Convergence of Some Random Series

Abstract

Let $\{ X_n \}$ be a sequence of independent random elements with values in a separable Banach space, $S_n = X_1 + \cdots + X_n$ and $\{a_n\}$ a sequence of nonnegative numbers. There is proved a criterion for the almost sure convergence of the series $\sum a_nS_n$. The special case $a_n = n^{–\alpha}, \alpha > 0$ is also studied.