A counterpart of the Borel-Cantelli lemma

Abstract

The general part of the Borel-Cantelli lemma says that for any sequence of events ( A n ) defined on a probability space (Ω, Σ, P ), the divergence of Σ n P ( A n ) is necessary for P ( A n i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the A n 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 ( A n ) 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.