Weak independence of events and the converse of the Borel–Cantelli Lemma

Csaba Biró; Israel R. Curbelo

doi:10.1016/j.exmath.2021.09.002

Weak independence of events and the converse of the Borel–Cantelli Lemma

Csaba Biró
, Israel R. Curbelo

Mathematics

Kean

Research output: Contribution to journal › Article › peer-review

Abstract

The converse of the Borel–Cantelli Lemma states that if {A_i}_i=1^∞ is a sequence of independent events such that ∑P(A_i)=∞, then almost surely infinitely many of these events will occur. Erdős and Rényi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.

Original language	English
Pages (from-to)	328-340
Number of pages	13
Journal	Expositiones Mathematicae
Volume	40
Issue number	2
DOIs	https://doi.org/10.1016/j.exmath.2021.09.002
State	Published - Jun 2022

ASJC Scopus subject areas

General Mathematics

Keywords

Borel–Cantelli Lemma
Independent events

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10.1016/j.exmath.2021.09.002

Cite this

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abstract = "The converse of the Borel–Cantelli Lemma states that if \{Ai\}i=1∞ is a sequence of independent events such that ∑P(Ai)=∞, then almost surely infinitely many of these events will occur. Erd{\H o}s and R{\'e}nyi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.",

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AU - Curbelo, Israel R.

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AB - The converse of the Borel–Cantelli Lemma states that if {Ai}i=1∞ is a sequence of independent events such that ∑P(Ai)=∞, then almost surely infinitely many of these events will occur. Erdős and Rényi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.

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KW - Independent events

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DO - 10.1016/j.exmath.2021.09.002

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JO - Expositiones Mathematicae

JF - Expositiones Mathematicae

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ER -

Weak independence of events and the converse of the Borel–Cantelli Lemma

Abstract

ASJC Scopus subject areas

Keywords

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