Open main menu

Wikipedia β

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.

Contents

Statement of lemma for probability spacesEdit

Let E1,E2,... be a sequence of events in some probability space. The Borel–Cantelli lemma states:[3]

If the sum of the probabilities of the En is finite
 
then the probability that infinitely many of them occur is 0, that is,
 

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

 

The theorem therefore asserts that if the sum of the probabilities of the events En 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.

ExampleEdit

Suppose (Xn) is a sequence of random variables with Pr(Xn = 0) = 1/n2 for each n. The probability that Xn = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [Xn = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ∑Pr(Xn = 0) converges to π2/6 ≈ 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 Xn = 0 occurring for infinitely many n is 0. Almost surely (i.e., with probability 1), Xn is nonzero for all but finitely many n.

Proof[4]Edit

Let   denote the indicator function of the event   (using Iverson bracket notation). Then, by the monotone convergence theorem

 

by hypothesis. This directly implies that

 

because otherwise

 

Alternative proof [5]Edit

Let (En) be a sequence of events in some probability space and suppose that the sum of the probabilities of the En is finite. That is suppose:

 

As the series converges, we must have that

 

Therefore :

 

Therefore it follows that

 

General measure spacesEdit

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

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

Converse resultEdit

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 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. That is:

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.

ExampleEdit

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 Ej is a collection of Lebesgue measurable subsets of a compact set in Rn such that

 

then there is a sequence Fj of translates

 

such that

 

apart from a set of measure zero.

Proof[5]Edit

Suppose that   and the events   are independent. It is sufficient to show the event that the En'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:

 

it is enough to show:  . Since the   are independent:

 

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  

CounterpartEdit

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.

See alsoEdit

ReferencesEdit

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

External linksEdit