Borel-Cantelli Lemma

Let  be a sequence of events occurring with a certain probability distribution, and let  be the event consisting of the occurrence of a finite number of events  for , 2, .... Then the probability of an infinite number of the  occurring is zero if

Equivalently, in the extreme case of  for all , the probability that none of them occurs is 1 and, in particular, the probability of  that a finite number occur is also 1.

REFERENCES:

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, pp. 435-436, 1988.