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

Presented by: 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, Volume 347 (2009) no. 21-22, pp. 1313-1316.

Abstract
Résumé

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})} .

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})} .

Received: 2009-06-20
Accepted: 2009-09-03
Published online: 2009-10-03

DOI: 10.1016/j.crma.2009.09.011

Author's affiliations:

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, Volume 347 (2009) no. 21-22, pp. 1313-1316. doi : 10.1016/j.crma.2009.09.011. http://www.numdam.org/articles/10.1016/j.crma.2009.09.011/

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