×

On the application of the Borel-Cantelli lemma. (English) Zbl 0046.35203

Let a sequence of events \(E_k\) be given and define \(\limsup E_k\) by \(\cap_{n=1}^\infty \cup_{k=n}^\infty E_k\). The authors state conditions for the probability of lim \(\limsup E_k\) to be unity. Their assumptions do not include independence of the events (as the Borel-Cantelli lemma does) and they are weaker than Borel’s condition \(\sum P (E_k \mid \bar E_1,...\bar E_{k-1})=\infty\). The result is applied to independent random variables which take the values \(\pm 1\) with probabilities \({1 \over 2}\).
Reviewer: St.Vajda

MSC:

60D05 Geometric probability and stochastic geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Borel, Traité du calcul des probabilités et de ses applications, vol. 2, no. 1, Applications à l’arithmétique et à la théorie des fonctions, Paris, Gauthier-Villars, 1926.
[2] William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons, Inc., New York, N.Y., 1950. · Zbl 0077.12201
[3] Kai-Lai Chung and Paul Erdös, On the lower limit of sums of independent random variables, Ann. of Math. (2) 48 (1947), 1003 – 1013. · Zbl 0029.15202 · doi:10.2307/1969391
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.