# Borel-Cantelli lemma

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2010 Mathematics Subject Classification: Primary: 60-01 Secondary: 60F1560F20 [MSN][ZBL]

A frequently used statement on infinite sequences of random events. Let $A_1,\dots, A_n, \dots$ be a sequence of events from a certain probability space and let $A$ be the event consisting in the occurance of (only) a finite number out of the events $A_n$, $n=1,2\dots$. Then, according to the Borel–Cantelli lemma, if

 (*)

then

If the events $A_n$ are mutually independent, then $\mathbb{P}(A) = 1$ 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.

#### References

 [B] E. Borel, "Les probabilités dénombrables et leurs applications arithmetiques" Rend. Circ. Mat. Palermo (2) , 27 (1909) pp. 247–271 Zbl 40.0283.01 [C] F.P. Cantelli, "Sulla probabilità come limite della frequenza" Atti Accad. Naz. Lincei , 26 : 1 (1917) pp. 39–45 Zbl 46.0779.02 [L] M. Loève, "Probability theory" , Princeton Univ. Press (1963) MR0203748 Zbl 0108.14202

#### Comments

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

#### References

 [F] W. Feller, "An introduction to probability theory and its applications", 1 , Wiley (1957) pp. Chapt.14
How to Cite This Entry:
Borel-Cantelli lemma. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Borel-Cantelli_lemma&oldid=33513
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article