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arXiv · 2602.23124

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

Istvan Berkes·Ioannis Karatzas·Walter Schachermayer
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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.

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Istvan Berkes, Ioannis Karatzas, Walter Schachermayer. 2026-02-26. Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers. https://arxiv.org/abs/2602.23124

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