On the Borel–Cantelli lemma and its generalization

Chunrong Feng; Liangpan Li; Jian Shen

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

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

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 :

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

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     author = {Chunrong Feng and Liangpan Li and Jian Shen},
     title = {On the {Borel{\textendash}Cantelli} lemma and its generalization},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1313--1316},
     publisher = {Elsevier},
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     year = {2009},
     doi = {10.1016/j.crma.2009.09.011},
     language = {en},
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Chunrong Feng; Liangpan Li; Jian Shen. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.011/

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