A Strong Law of Large Numbers for Subsequences of Random Elements in Separable Banach Spaces

Abstract

In 1967, Komlos proved that if $\{\xi_n\}$ is a sequence of real random variables for which $\sup_{n \geqslant 1}E|\xi_n| < \infty$, then there exists a subsequence $\{\eta_n\}$ of $\{\xi_n\}$ and an integrable random variable $\eta$ such that for an arbitrary subsequence $\{\eta_n\}$ of $\{\eta_n\}$. $$\lim_{n \rightarrow \infty} \frac{1}{n}(\check{\eta}_1 + \check{\eta}_2 + \cdots + \check{\eta}_n) = \eta \mathrm{a.s.}$$ In this paper, we attempt to extend this result to separable Banach space valued random elements. We impose a condition stronger than uniform integrability.