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Borel

Borel

[baw-rel, buh-; Fr. baw-rel]
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).
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.
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.

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

If the sum of the probabilities of the En is finite

sum_{n=1}^infty P(E_n)

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 (Xn) is a sequence of random variables, with Pr(Xn = 0) = 1/n2 for each n. The sum of Pr(Xn = 0) is finite (in fact it is pi^2/6 - see Riemann zeta function), so the Borel-Cantelli Lemma says that the probability of Xn = 0 occurring for infinitely many n is 0. In other words Xn is nonzero almost surely for all but finitely many n.

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

Let μ be a measure on a set X, with σ-algebra F, and let (An) be a sequence in F. If

sum_{n=1}^inftymu(A_n)

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 En are independent and the sum of the probabilities of the En 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 Rn. Specifically , if Ej is a collection of Lebesgue measurable subsets of a compact set in Rn such that

sum_j mu(E_j) = infty,

then there is a sequence Fj 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.

Counterpart

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.

References

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