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Borel, Félix Édouard Émile, 1871-1956, French mathematician. He is noted for his work in infinitesimal calculus and the calculus of probabilities. He was professor at the Univ. of Paris (1904-41), director of the Henri Poincaré Institute (from 1927), and a representative in the French chamber of deputies (1924-36).

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

Borel, Petrus, pseud. of **Joseph-Pierre Borel D'Hauterive,** 1809-59, French novelist, poet, and translator. Although trained as an architect, he soon turned to writing. Borel was the most extreme of the *bousingos,* a group of extravagant young romantic artists and writers. He loathed the bourgeoisie and believed in the hatred of men for each other. Among his works, whose aim was to shock, are *Rhapsodies* (1832) and *Madame Putip-her* (1839), both of which are horrifying and melodramic.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

In probability theory, the Borel-Cantelli lemma is a theorem about sequences of events. In a slightly more general form, it is also a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli.## Counterpart

## References

## External links

Let (E_{n}) be a sequence of events in some probability space.
The Borel-Cantelli lemma states:

- If the sum of the probabilities of the E
_{n}is finite

- $sum\_\{n=1\}^infty\; P(E\_n),\; math>$

- then the probability that infinitely many of them occur is 0, that is,

- $Pleft(limsup\_\{ntoinfty\}\; E\_nright)\; =\; 0.,$

Here, "lim sup" denotes limit superior of the events considered as sets. Note that no assumption of independence is required.

For example, suppose (X_{n}) is a sequence of random variables, with Pr(X_{n} = 0) = 1/n^{2} for each n. The sum of Pr(X_{n} = 0) is finite (in fact it is $pi^2/6$ - see Riemann zeta function), so the Borel-Cantelli Lemma says that the probability of X_{n} = 0 occurring for infinitely many n is 0. In other words X_{n} is nonzero almost surely for all but finitely many n.

For general measure spaces, the Borel-Cantelli lemma takes the following form:

- $sum\_\{n=1\}^inftymu(A\_n),\; math>$

- then

- $muleft(limsup\_\{ntoinfty\}\; A\_nright)\; =\; 0.,$

A related result, sometimes called the second Borel-Cantelli lemma, is a partial converse of the first Borel-Cantelli lemma. It says:

- If the events E
_{n}are independent and the sum of the probabilities of the E_{n}diverges to infinity, then the probability that infinitely many of them occur is 1.

The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult.

The infinite monkey theorem is a special case of this lemma.

The lemma can be applied to give a covering theorem in R^{n}. Specifically , if E_{j} is a collection of Lebesgue measurable subsets of a compact set in R^{n} such that

- $sum\_j\; mu(E\_j)\; =\; infty,$

then there is a sequence F_{j} of translates

- $F\_j\; =\; E\_j\; +\; x\_j$

such that

- $limsup\; F\_j\; =\; bigcap\_\{n=1\}^infty\; bigcup\_\{k=n\}^infty\; F\_k\; =\; mathbb\{R\}^n$

apart from a set of measure zero.

Another related result is the so-called counterpart of the Borel-Cantelli lemma. It is a counterpart of the Lemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that $,(A\_n\; )$ is monotone increasing for sufficiently large indices. This Lemma says:

Let $,(A\_n\; ),$ be such that $A\_\{k\}\; subseteq\; A\_\{k+1\}$, and let $,bar\; A\; ,$ denote the complement of $,\; A\; ,$.

Then the probability of infinitely many $,\; A\_k\; ,$ occur (that is, at least one $,\; A\_k\; ,$ occurs) is one if and only if there exists a strictly increasing sequence of positive integers $,(t\_\; k\; ),$ such that

- $sum\_\{k\}\; P(A\_\{t\_\{k+1\}\}|\; bar\; A\_\{t\_k\})\; =\; infty.$

This simple result can be useful in problems such as for instance those involving hitting probabilities for stochastic process with the choice of the sequence $,(t\_\; k\; ),$ usually being the essence.

- .
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- Planet Math Proof Refer for a simple proof of the Borel Cantelli Lemma

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Last updated on Sunday September 28, 2008 at 06:50:23 PDT (GMT -0700)

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