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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
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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. · JFM 52.0525.05
[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
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