Hereditary Hsu-Robbins-Erdös Law of Large Numbers
Istvan Berkes, Ioannis Karatzas, Walter Schachermayer
TL;DR
This paper establishes a hereditary enhancement of the Hsu–Robbins–Erdős law of large numbers: for any sequence $(f_n)$ with uniformly bounded second moments, there exists a subsequence whose Cesàro means converge completely to some $f_\infty\in\mathbb{L}^2$, and this complete convergence persists along all further subsequences. The authors split the result into a sufficiency part and a necessity part, employing a blend of martingale techniques, exchangeability at infinity, and KPR-type decompositions, with a key auxiliary result requiring a star-condition on a weak-$\mathbb{L}^1$ limit of squared terms. A central methodological advance is the construction of an omnibus, progressively exchangeable sequence that approximates a subsequence in $\mathbb{L}^2$, controlled via Wasserstein distances and aggregation of conditional laws, enabling hereditary complete convergence. The work connects to de Finetti-type representations for exchangeable sequences, Komlós subsequence principles, and quantitative HR–E bounds, and shows that hereditary complete convergence can necessitate exchangeability-based approximations beyond martingale-difference techniques. The results deepen our understanding of subsequence principles in probability, with potential implications for ergodic-type convergence under weak dependence structures.
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{Cesàro} mean to some $\,f_\infty \in \mathbb{L}^2$ {\it completely,} to wit, $ \sum_{N \in \N} \, ¶\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{Cesàro} mean, but necessary as well.