Borel-Cantelli lemma.html

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 (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

$\sum_{n=1}^\infty P(E_n)<\infty,$

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

$P\left(\limsup_{n\to\infty} E_n\right) = 0.\,$

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/n² for each n. The sum of Pr(X_n = 0) is finite (in fact it is $π 2 / 6$ - 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:

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

$\sum_{n=1}^\infty\mu(A_n)<\infty,$

then

$\mu\left(\limsup_{n\to\infty} A_n\right) = 0.\,$

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 Rⁿ. Specifically (Stein 1993, Lemma X.2.1), if E_j 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 F_j of translates

F j = E j + x j

such that

$\lim\sup F_j = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty F_k = \mathbb{R}^n$

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

Feller, William (1961), An Introduction to Probability Theory and Its Application, John Wiley & Sons .
Stein, Elias (1993), Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press .
Bruss, F. Thomas (1980), "A counterpart of the Borel Cantelli Lemma", J. Appl. Prob. 17: 1094-1101 .

External links

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