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A counterpart of the Borel-Cantelli lemma

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss
F. Thomas Bruss*
Affiliation:
Facultés Universitaires de Namur
*
∗Postal address: Facultés Universitaires de Namur, Département de Mathématiques, Rue de Bruxelles 61, B-5000 Namur, Belgium.
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Abstract

The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ, P), the divergence of ΣnP(An) is necessary for P(An i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the An are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by for some . We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory of ϕ -branching processes will show the tractability of the method.

Keywords

BOREL-CANTELLI LEMMARANDOM WALKϕ-BRANCHING PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 1094 - 1101
Copyright
Copyright © Applied Probability Trust 

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References

[1] Breiman, L. (1968) Probability. Addison-Wesley, Reading, MA.Google Scholar
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