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In probability theory, the Borel–Cantelli
lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...
is a theorem about sequences of events. In general, it is a result in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
. It is named after Émile Borel and
Francesco Paolo Cantelli Francesco Paolo Cantelli (20 December 187521 July 1966) was an Italian mathematician. He made contributions to celestial mechanics, probability theory, and actuarial science. Biography Cantelli was born in Palermo. He received his doctorate in ...
, who gave statement to the lemma in the first decades of the 20th century. 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: 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, \limsup_ E_n = \bigcap_^\infty \bigcup_^\infty E_k. The set lim sup ''E''''n'' is sometimes denoted , 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 variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s 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'' = 0events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ΣPr(''X''''n'' = 0) converges to 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 \left\^\infty_ is non-increasing: \bigcup_^\infty E_n \supseteq \bigcup_^\infty E_n \supseteq \cdots \supseteq \bigcup_^\infty E_n \supseteq \bigcup_^\infty E_n \supseteq \cdots \supseteq \limsup_ E_n. By continuity from above, \Pr(\limsup_ E_n) = \lim_\Pr\left(\bigcup_^\infty E_n\right). By subadditivity, \Pr\left(\bigcup_^\infty E_n\right) \leq \sum^\infty_ \Pr(E_n). By original assumption, \sum_^\infty \Pr(E_n)<\infty. As the series \sum_^\infty \Pr(E_n) converges, \lim_ \sum^\infty_ \Pr(E_n)=0, as required.


General measure spaces

For general
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
s, the Borel–Cantelli lemma takes the following form:


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: 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 , 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 \lim\sup F_j = \bigcap_^\infty \bigcup_^\infty F_k = \mathbb^n apart from a set of measure zero.


Proof

Suppose that \sum_^\infty \Pr(E_n) = \infty and the events (E_n)^\infty_ 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 1-\Pr(\limsup_ E_n) = 0. Noting that: \begin 1 - \Pr(\limsup_ E_n) &= 1 - \Pr\left(\\right) = \Pr\left(\^c \right) \\ & = \Pr\left(\left(\bigcap_^\infty \bigcup_^\infty E_n\right)^c \right) = \Pr\left(\bigcup_^\infty \bigcap_^\infty E_n^c \right)\\ &= \Pr\left(\liminf_E_n^\right)= \lim_\Pr\left(\bigcap_^\infty E_n^c \right) \end , it is enough to show: \Pr\left(\bigcap_^ E_n^\right) = 0. Since the (E_n)^_ are independent: \begin \Pr\left(\bigcap_^\infty E_n^c\right) &= \prod^\infty_ \Pr(E_n^c) \\ &= \prod^\infty_ (1-\Pr(E_n)). \end The
convergence test In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n. List of tests Limit of the summand If th ...
for
infinite products In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
guarantees that the product above is 0, if \sum_^\infty \Pr(E_n) diverges. This completes the proof.


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_, 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 \Pr( A_ \mid \bar A_) = \infty. This simple result can be useful in problems such as for instance those involving hitting probabilities for
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
with the choice of the sequence (t_k) usually being the essence.


Kochen–Stone

Let A_n be a sequence of events with \sum\Pr(A_n)=\infty and \liminf_ \frac < \infty, then there is a positive probability that A_n occur infinitely often.


See also

* Lévy's zero–one law * Kuratowski convergence * Infinite monkey theorem


References

* * . * . * . * 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 {{DEFAULTSORT:Borel-Cantelli lemma Theorems in measure theory Probability theorems Covering lemmas Lemmas