A Strong Law of Large Numbers for Positive Random Variables

TL;DR

This paper proves a new strong law of large numbers for nonnegative random variables, showing subsequence convergence in Cesàro mean, using elementary methods that strengthen previous results by replacing convex combinations.

Contribution

Introduces a novel elementary proof of a strong law of large numbers for positive variables, improving prior theorems by emphasizing Cesàro means and subsequence convergence.

Findings

Subsequences converge almost everywhere in Cesàro mean

Methodology simplifies proof using elementary tools

Strengthens previous theorems by replacing convex combinations

Abstract

In the spirit of the famous KOML\'OS (1967) theorem, every sequence of nonnegative, measurable functions $\{ f_n \}_{n \in \N}$ on a probability space, contains a subsequence which - along with all its subsequences - converges a.e. in CES\`ARO mean to some measurable $f_* : \Omega \to [0, \infty]$. This result of VON WEIZS\"ACKER (2004) is proved here using a new methodology and elementary tools; these sharpen also a theorem of DELBAEN & SCHACHERMAYER (1994), replacing general convex combinations by CES\`ARO means.