The Divergence Borel-Cantelli Lemma revisited

TL;DR

This paper revisits the divergence Borel-Cantelli Lemma, analyzing the conditions under which the limit superior set of events has positive or full measure, especially focusing on independence hypotheses.

Contribution

It establishes both necessary and sufficient conditions for the measure of the limit superior set under various independence assumptions.

Findings

Identifies conditions for $E_ extinfty$ to have positive measure.

Provides criteria for $E_ extinfty$ to have full measure.

Clarifies the role of independence in divergence Borel-Cantelli results.

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.