Borel-Cantelli Lemma
المؤلف:
Hazewinkel, M. (Managing Ed.).
المصدر:
Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel
الجزء والصفحة:
pp. 435-436
7-3-2021
2137
Borel-Cantelli Lemma
Let ![<span style=]()
{A_n}_(n=0)^infty" src="https://mathworld.wolfram.com/images/equations/Borel-CantelliLemma/Inline1.gif" style="height:17px; width:44px" /> 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.
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