The divergence Borel–Cantelli Lemma revisited

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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.
Original languageEnglish
Article number126750
Number of pages21
JournalJournal of mathematical analysis and applications
Issue number1
Early online date13 Oct 2022
Publication statusPublished - 1 Mar 2023

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