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A related result, sometimes called the second Borel-Cantelli lemma, is a partial converse of the first Borel-Cantelli lemma. It says:
:If the events "E""n" are independent and the sum of the probabilities of the "E""n" diverges to infinity, then the probability that infinitely many of them occur is 1.
The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult.
The infinite monkey theorem is a special case of this lemma.
The lemma can be applied to give a covering theorem in R"n". Specifically harv|Stein|1993|loc=Lemma X.2.1, if "E""j" is a collection of Lebesgue measurable subsets of a compact set in R"n" such that
:
then there is a sequence "F""j" of translates
:
such that
:
apart from a set of measure zero.
Counterpart
Another related result is the so-called counterpart of the Borel-Cantelli lemma. It is a counterpart of theLemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that is monotone increasing for sufficiently large indices. This Lemma says:
Let be such that ,and let denote the complement of .
Then the probability of infinitely many occur (that is, at least one occurs) is one if and only ifthere exists a strictly increasing sequence of positive integers such that
:
This simple result can be useful in problems such as for instance those involving hitting probabilities for stochastic process with the choice of the sequence usually being the essence.
References
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External links
* [http://planetmath.org/encyclopedia/BorelCantelliLemma.html Planet Math Proof] Refer for a simple proof of the Borel Cantelli Lemma