# Borel–Cantelli lemma

*https://en.wikipedia.org/wiki/Borel-Cantelli_lemma*

In
probability theory, the **Borel–Cantelli
lemma** is a
theorem about
sequences of
events. In general, it is a result in
measure theory. It is named after
Émile Borel and
Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century.^{
[1]}^{
[2]} A related result, sometimes called the **second Borel–Cantelli lemma**, is a partial
converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have probability of either zero or one. Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include
Kolmogorov's zero–one law and the
Hewitt–Savage zero–one law.

## Statement of lemma for probability spaces

Let *E*_{1},*E*_{2},... be a sequence of events in some
probability space.
The Borel–Cantelli lemma states:^{
[3]}

**Borel–Cantelli lemma** — If the sum of the probabilities of the events {*E*_{n}} is finite

Here, "lim sup" denotes
limit supremum of the sequence of events, and each event is a set of outcomes. That is, lim sup *E*_{n} is the set of outcomes that occur infinitely many times within the infinite sequence of events (*E*_{n}). Explicitly,

The set lim sup *E*_{n} is sometimes denoted {*E*_{n} i.o. }, where "i.o." stands for "infinitely often". The theorem therefore asserts that if the sum of the probabilities of the events *E*_{n} is finite, then the set of all outcomes that are "repeated" infinitely many times must occur with probability zero. Note that no assumption of
independence is required.

### Example

Suppose (*X*_{n}) is a sequence of
random variables with Pr(*X*_{n} = 0) = 1/*n*^{2} for each *n*. The probability that *X*_{n} = 0 occurs for infinitely many *n* is equivalent to the probability of the intersection of infinitely many [*X*_{n} = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ∑Pr(*X*_{n} = 0) converges to π^{2}/6 ≈ 1.645 < ∞, and so the Borel–Cantelli Lemma states that the set of outcomes that are common to infinitely many such events occurs with probability zero. Hence, the probability of *X*_{n} = 0 occurring for infinitely many *n* is 0.
Almost surely (i.e., with probability 1), *X*_{n} is nonzero for all but finitely many *n*.

## Proof

Let (*E*_{n}) be a sequence of events in some
probability space.

The sequence of events is non-increasing:

By continuity from above,

By subadditivity,

By original assumption, As the series converges,

^{ [4]}

## General measure spaces

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

**Borel–Cantelli Lemma for measure spaces** — Let *μ* be a (positive)
measure on a set *X*, with
σ-algebra *F*, and let (*A*_{n}) be a sequence in *F*. If

## Converse result

A related result, sometimes called the **second Borel–Cantelli lemma**, is a partial converse of the first Borel–Cantelli lemma. The lemma states: 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. That is:

**Second Borel–Cantelli Lemma** — If and the events are independent, then

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

### Example

The infinite monkey theorem, that endless typing at random will, with probability 1, eventually produce every finite text (such as the works of Shakespeare), amounts to the statement that a (not necessarily fair) coin tossed infinitely often will eventually come up Heads. This is a special case of the second Lemma.

The lemma can be applied to give a covering theorem in **R**^{n}. Specifically (
Stein 1993, Lemma X.2.1), if *E*_{j} is a collection of
Lebesgue measurable subsets of a
compact set in **R**^{n} such that

*F*

_{j}of translates

## Proof

Suppose that and the events are independent. It is sufficient to show the event that the *E*_{n}'s did not occur for infinitely many values of *n* has probability 0. This is just to say that it is sufficient to show that

Noting that:

This completes the proof. Alternatively, we can see by taking negative the logarithm of both sides to get:

Since −log(1 − *x*) ≥ *x* for all *x* > 0, the result similarly follows from our assumption that ^{
[4]}

## 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 is monotone increasing for sufficiently large indices. This Lemma says:

Let be such that , and let denote the complement of . Then the probability of infinitely many occur (that is, at least one occurs) is one if and only if there exists a strictly increasing sequence of positive integers such that

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 usually being the essence.

## Kochen–Stone

Let be a sequence of events with and then there is a positive probability that occur infinitely often.

## See also

## References

This article includes a list of general
references, but it remains largely unverified because it lacks sufficient corresponding
inline citations. (November 2009) |

**^**E. Borel, "Les probabilités dénombrables et leurs applications arithmetiques"*Rend. Circ. Mat. Palermo*(2)**27**(1909) pp. 247–271.**^**F.P. Cantelli, "Sulla probabilità come limite della frequenza",*Atti Accad. Naz. Lincei*26:1 (1917) pp.39–45.**^**Klenke, Achim (2006).*Probability Theory*. Springer-Verlag. ISBN 978-1-84800-047-6.- ^
^{a}^{b}"Romik, Dan. Probability Theory Lecture Notes, Fall 2009, UC Davis" (PDF). Archived from the original (PDF) on 2010-06-14.

- Prokhorov, A.V. (2001) [1994],
"Borel–Cantelli lemma",
*Encyclopedia of Mathematics*, EMS Press -
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. Probab.*,**17**: 1094–1101. - Durrett, Rick. "Probability: Theory and Examples." Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005.

## External links

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