Hereditary Hsu-Robbins-Erd\"os Law of Large Numbers

Abstract

We show that every sequence $f_1, f_2, \cdots$ of real-valued random variables with $\sup_{n \in \N} \E (f_n^2) < \infty$ contains a subsequence $f_{k_1}, f_{k_2}, \cdots$ converging in \textsc{Ces\`aro} mean to some $\,f_\infty \in \mathbb{L}^2$ {\it completely,} to wit, $ \sum_{N \in \N} \, \P \left( \bigg| \frac{1}{N} \sum_{n=1}^N f_{k_n} - f_\infty \bigg| > \eps \right)< \infty\,, \quad \forall ~ \eps > 0\,; $ and {\it hereditarily,} i.e., along all further subsequences as well. We also identify a condition, slightly weaker than boundedness in $ \mathbb{L}^2,$ which turns out to be not only sufficient for the above hereditary complete convergence in \textsc{Ces\`aro} mean, but necessary as well.