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,
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
''n'' = 0">'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
is non-increasing:
By continuity from above,
By subadditivity,
By original assumption,
As the series
converges,
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
then there is a sequence ''F''
''j'' of translates
such that
apart from a set of measure zero.
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:
, it is enough to show:
. Since the
are independent:
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
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
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
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
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
*
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 ProofRefer for a simple proof of the Borel Cantelli Lemma
{{DEFAULTSORT:Borel-Cantelli lemma
Theorems in measure theory
Probability theorems
Covering lemmas
Lemmas