On the Borel–Cantelli lemma and its generalization

Feng, Chunrong; Li, Liangpan; Shen, Jian

doi:10.1016/j.crma.2009.09.011

Probability Theory

On the Borel–Cantelli lemma and its generalization
[Sur le lemme de Borel–Cantelli et sa généralisation]

Présenté par : Marc Yor

Feng, Chunrong ^1,² ; Li, Liangpan ^1,³ ; Shen, Jian ³

¹ Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
² Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK
³ Department of Mathematics, Texas State University, San Marcos, TX 78666, USA

Comptes Rendus. Mathématique, Tome 347 (2009) no. 21-22, pp. 1313-1316.

Résumé
Abstract

Soit ${A_{n}}_{n = 1}^{\infty}$ une séquence d'événements dans un éspace de probabilité $(Ω, F, P)$ . On montre que, si $\lim_{m \to \infty} \sum_{n = 1}^{m} w_{n} P (A_{n}) = \infty$ où chaque $w_{n} \in R$ , alors

P (lim sup A_{n}) ⩾ \underset{n \to \infty}{lim sup} \frac{(\sum_{k = 1}^{n} w_{k} P {(A_{k}))}^{2}}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} P (A_{i} \cap A_{j})} .

Let ${A_{n}}_{n = 1}^{\infty}$ be a sequence of events on a probability space $(Ω, F, P)$ . We show that if $\lim_{m \to \infty} \sum_{n = 1}^{m} w_{n} P (A_{n}) = \infty$ where each $w_{n} \in R$ , then

P (lim sup A_{n}) ⩾ \underset{n \to \infty}{lim sup} \frac{(\sum_{k = 1}^{n} w_{k} P {(A_{k}))}^{2}}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} P (A_{i} \cap A_{j})} .

Reçu le : 2009-06-20
Accepté le : 2009-09-03
Publié le : 2009-10-03

DOI : 10.1016/j.crma.2009.09.011

Affiliations des auteurs :

Feng, Chunrong ^{1, 2} ; Li, Liangpan ^{1, 3} ; Shen, Jian ³

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     title = {On the {Borel{\textendash}Cantelli} lemma and its generalization},
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Feng, Chunrong; Li, Liangpan; Shen, Jian. On the Borel–Cantelli lemma and its generalization. Comptes Rendus. Mathématique, Tome 347 (2009) no. 21-22, pp. 1313-1316. doi : 10.1016/j.crma.2009.09.011. http://archive.numdam.org/articles/10.1016/j.crma.2009.09.011/

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