# Borel-Cantelli lemma

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2020 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 occurrence of (only) a finite number out of the events $A_n$, $n=1,2\dots$. Then, according to the Borel–Cantelli lemma, if

$$\label{eq1} \sum\limits_{n=1}^{\infty}\mathbb P(A_n) < \infty$$

then

$$\mathbb P(A) = 1.$$

If the events $A_n$ are mutually independent, then $\mathbb{P}(A) = 1$ or $0$, depending on whether the series $\sum\limits_{n=1}^{\infty}\mathbb P(A_n)$ converges or diverges, i.e. in this case the condition \eqref{eq1} is necessary and sufficient for $\mathbb P(A) = 1$; 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