Necessary and Sufficient Conditions for the Lacunary/Hereditary 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.