Borel–Cantelli lemma

Theorem in probability

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 E₁,E₂,... be a sequence of events in some probability space. The Borel–Cantelli lemma states:^[3]^[4]

Borel–Cantelli lemma — If the sum of the probabilities of the events {E_n} is finite

\sum _{n=1}^{\infty }\Pr(E_{n})<\infty ,

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

\Pr \left(\limsup _{n\to \infty }E_{n}\right)=0.

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 _{n\to \infty }E_{n}=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }E_{k}.

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/n² 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 ΣPr(X_n = 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 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 ${\textstyle \left\{\bigcup _{n=N}^{\infty }E_{n}\right\}_{N=1}^{\infty }}$ is non-increasing:

\bigcup _{n=1}^{\infty }E_{n}\supseteq \bigcup _{n=2}^{\infty }E_{n}\supseteq \cdots \supseteq \bigcup _{n=N}^{\infty }E_{n}\supseteq \bigcup _{n=N+1}^{\infty }E_{n}\supseteq \cdots \supseteq \limsup _{n\to \infty }E_{n}.

By continuity from above,

\Pr(\limsup _{n\to \infty }E_{n})=\lim _{N\to \infty }\Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right).

By subadditivity,

\Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right)\leq \sum _{n=N}^{\infty }\Pr(E_{n}).

By original assumption, ${\textstyle \sum _{n=1}^{\infty }\Pr(E_{n})<\infty .}$ As the series ${\textstyle \sum _{n=1}^{\infty }\Pr(E_{n})}$ converges,

\lim _{N\to \infty }\sum _{n=N}^{\infty }\Pr(E_{n})=0,

as required.^[5]

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

\sum _{n=1}^{\infty }\mu (A_{n})<\infty ,

then

\mu \left(\limsup _{n\to \infty }A_{n}\right)=0.

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:^[4]

Second Borel–Cantelli Lemma — If $\sum _{n=1}^{\infty }\Pr(E_{n})=\infty$ and the events $(E_{n})_{n=1}^{\infty }$ are independent, then $\Pr(\limsup _{n\to \infty }E_{n})=1.$

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

The infinite monkey theorem follows from the Second lemma.

Example

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.

Proof

Suppose that ${\textstyle \sum _{n=1}^{\infty }\Pr(E_{n})=\infty }$ and the events $(E_{n})_{n=1}^{\infty }$ 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 _{n\to \infty }E_{n})=0.

Noting that:

{\begin{aligned}1-\Pr(\limsup _{n\to \infty }E_{n})&=1-\Pr \left(\{E_{n}{\text{ i.o.}}\}\right)=\Pr \left(\{E_{n}{\text{ i.o.}}\}^{c}\right)\\&=\Pr \left(\left(\bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }E_{n}\right)^{c}\right)=\Pr \left(\bigcup _{N=1}^{\infty }\bigcap _{n=N}^{\infty }E_{n}^{c}\right)\\&=\Pr \left(\liminf _{n\to \infty }E_{n}^{c}\right)=\lim _{N\to \infty }\Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right),\end{aligned}}

it is enough to show:

{\textstyle \Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right)=0}

. Since the

(E_{n})_{n=1}^{\infty }

are independent:

{\begin{aligned}\Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right)&=\prod _{n=N}^{\infty }\Pr(E_{n}^{c})\\&=\prod _{n=N}^{\infty }(1-\Pr(E_{n})).\end{aligned}}

The convergence test for infinite products guarantees that the product above is 0, if

{\textstyle \sum _{n=N}^{\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 $A_{k}\subseteq A_{k+1}$ , 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}\Pr(A_{t_{k+1}}\mid {\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.

Kochen–Stone

Let $(A_{n})$ be a sequence of events with ${\textstyle \sum \Pr(A_{n})=\infty }$ and ${\textstyle \liminf _{k\to \infty }{\frac {\sum _{1\leq m,n\leq k}\Pr(A_{m}\cap A_{n})}{\left(\sum _{n=1}^{k}\Pr(A_{n})\right)^{2}}}<\infty .}$ Then there is a positive probability that $A_{n}$ occur infinitely often.

References

^ 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 Shiryaev, Albert N. (2016). Probability-1: Volume 1. Graduate Texts in Mathematics. Vol. 95. New York, NY: Springer New York. doi:10.1007/978-0-387-72206-1. ISBN 978-0-387-72205-4.
^ "Romik, Dan. Probability Theory Lecture Notes, Fall 2009, UC Davis" (PDF). Archived from the original (PDF) on 2010-06-14.

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