On the Almost Sure Convergence of Randomly Weighted Sums of Random Elements

Abstract

Let $\{X_n\}$ be random elements in a separable Banach space which is $p$-smoothable and let $\{a_k\}$ and $\{A_k\}$ denote positive random variables such that almost surely $A_k$ is monotonically increasing to $\infty$ and that $A_k/a_k \rightarrow \infty$. Convergence almost surely is obtained for the weighted sum $A^{-1}_n \sum^n_{k=1} a_kX_k$ and is related to a moment condition on the random elements and a growth condition on the random weights.