Almost Sure and Complete Convergence of Randomly Weighted Sums of Independent Random Elements in Banach Spaces

Abstract

This work develops almost sure and complete convergence of randomly weighted sums of independent random elements in a real separable Banach space. Sufficient conditions for almost sure convergence and complete convergence in the sense defined by Hsu and Robbins are provided. Examples showing that the conditions cannot be removed or weakened are given. It is also demonstrated that some of the known theorems in the literature are special cases of our results.