Borel-Cantelli lemma

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A frequently used statement on infinite sequences of random events. Let be a sequence of events from a certain probability space and let be the event consisting in the occurance of (only) a finite number out of the events , . Then, according to the Borel–Cantelli lemma, if



If the events are mutually independent, then or 0, depending on whether the series converges or diverges, i.e. in this case the condition (*) is necessary and sufficient for ; this is the so-called Borel criterion for "zero or one" (cf. Zero-one law). This last criterion can be generalized to include certain classes of dependent events. The Borel–Cantelli lemma is used, for example, to prove the strong law of large numbers.


[1] E. Borel, "Les probabilités dénombrables et leurs applications arithmetique" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271
[2] F.P. Cantelli, "Sulla probabilità come limite della frequenza" Atti Accad. Naz. Lincei , 26 : 1 (1917) pp. 39–45
[3] M. Loève, "Probability theory" , Princeton Univ. Press (1963)


The Borel–Cantelli lemma can be used in number theory to prove the so-called "normality" of almost-all natural numbers, cf. [a1], Chapt. 8, Sect. 6.


[a1] W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1957) pp. Chapt.14
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Borel-Cantelli lemma. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article