Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers

TL;DR

This paper establishes necessary and sufficient conditions for lacunary and hereditary laws of large numbers, extending classical results and addressing open questions for general and exchangeable sequences.

Contribution

It identifies weaker, Egorov-type conditions as both necessary and sufficient for lacunary/hereditary laws of large numbers, generalizing prior theorems.

Findings

Egorov-type conditions are necessary and sufficient

Conditions developed for exchangeable sequences

Addresses open questions in the theory of laws of large numbers

Abstract

The celebrated theorem of Komlos asserts that L1-boundedness is sufficient for a given sequence of functions to contain a subsequence along which (in a "lacunary" manner), and along whose every further subsequence ("hereditarily"), a strong law of large numbers holds. We identify here slightly weaker, Egorov-type conditions, as not only sufficient in this context, but necessary as well. Necessary and sufficient conditions are developed also for the lacunary/hereditary version of the weak law of large numbers for general sequences, as well as for the weak law of large numbers in the context of exchangeable sequences, both long-open questions.