Beresnevich, Victor orcid.org/0000-0002-1811-9697 and Velani, Sanju orcid.org/0000-0002-4442-6316 (2023) The divergence Borel–Cantelli Lemma revisited. Journal of mathematical analysis and applications. 126750. ISSN 0022-247X
Abstract
Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The classical Borel--Cantelli Lemma states that for any sequence of $\mu$-measurable sets $E_i$ ($i=1,2,3,\dots$), if the sum of their measures converges then the corresponding $\limsup$ set $E_\infty$ is of measure zero. In general the converse statement is false. However, it is well known that the divergence counterpart is true under various additional `independence' hypotheses. In this paper we revisit these hypotheses and establish both sufficient and necessary conditions for $E_\infty$ to have either positive or full measure.
Metadata
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Copyright, Publisher and Additional Information: | © 2022 The Author(s). | ||||
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Institution: | The University of York | ||||
Academic Units: | The University of York > Faculty of Sciences (York) > Mathematics (York) | ||||
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Depositing User: | Pure (York) | ||||
Date Deposited: | 06 Oct 2022 14:00 | ||||
Last Modified: | 06 Dec 2023 14:54 | ||||
Published Version: | https://doi.org/10.1016/j.jmaa.2022.126750 | ||||
Status: | Published | ||||
Refereed: | Yes | ||||
Identification Number: | https://doi.org/10.1016/j.jmaa.2022.126750 |
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