A counterpart of the Borel-Cantelli lemma
Author:
- F. Thomas BrussFBF. Thomas Bruss
- 1 December 1980
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 17 (4) , 1094-1101
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
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