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Borel-Cantelli Lemma

Let $\{A_n\}_{n=0}^\infty$ be a Sequence of events occurring with a certain probability distribution, and let $A$ be the event consisting of the occurrence of a finite number of events $A_n$, $n=1$, .... Then if

\begin{displaymath}
\sum_{n=1}^\infty P(A_n)<\infty,
\end{displaymath}

then

\begin{displaymath}
P(A)=1.
\end{displaymath}


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.




© 1996-9 Eric W. Weisstein
1999-05-26