The weakly dependent strong law of large numbers revisited

TL;DR

This paper presents a simple, elementary proof of the strong law of large numbers under a specific decay condition, also providing a convergence rate without relying on maximal inequalities.

Contribution

It offers a novel, self-contained proof of the strong law under power law decay, inspired by quantum field theory methods, and includes a convergence rate.

Findings

Elementary proof of the strong law under power law decay

Provides a convergence rate for the law

Avoids maximal inequalities in the proof

Abstract

We give a short, self-contained, and elementary proof of the strong law of large numbers under a power law decay hypothesis for joint second moments. The result is related to the classical one by Lyons. However, we also provide a rate of convergence. Our proof does not use maximal inequalities and is instead inspired by the method of multiscale large versus small field decompositions in constructive quantum field theory.