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On the First and the Second Borel-Cantelli Lemmas

Jian-Sheng Xie, Qihang Wang

Abstract

Let $\{A_n\}_{n=1}^\infty$ be a sequence of events and let $\displaystyle S:=\sum_{n=1}^\infty 1_{A_n}$. We present in this note equivalent characterizations for the statements $\mathbb{P} (S<\infty)=1$ and $\mathbb{P} (S=\infty)=1$ respectively. These characterizations are of Borel-Cantelli lemma type and of Kochen-Stone lemma type respectively, which could be regarded as the most general version of the first and the second Borel-Cantelli Lemmas.

On the First and the Second Borel-Cantelli Lemmas

Abstract

Let be a sequence of events and let . We present in this note equivalent characterizations for the statements and respectively. These characterizations are of Borel-Cantelli lemma type and of Kochen-Stone lemma type respectively, which could be regarded as the most general version of the first and the second Borel-Cantelli Lemmas.
Paper Structure (3 sections, 6 theorems, 24 equations)

This paper contains 3 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['thm: main-A']} and \ref{['thm: main-B']}
  3. Discussions

Key Result

Lemma 1.1

(Borel-Cantelli Lemma) If a sequence $\{A_n\}_{n=1}^\infty$ of events satisfies then one has

Theorems & Definitions (6)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6