The Borel–Cantelli lemma is a result in measure theory. It is often stated in the context of probability theory, where it is used to study whether, in a given sequence of events, a finite or infinite number of these events occur. The statement of the lemma is often split into two parts:
The first Borel–Cantelli lemma, which states that if the sum of the probabilities of the events is finite, then the probability that infinitely many of them occur is 0. This result holds for any sequence of events, without additional assumptions; The second Borel–Cantelli lemma, which states that if the events are independent and the sum of their probabilities is infinite, then the probability that infinitely many of them occur is 1. It follows that the probability of the limit superior of a sequence of independent events is always either zero or one. For this reason, the Borel–Cantelli lemma is often referred to as a zero-one law. Other examples or similar results include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law. The Borel–Cantelli lemma is named after Émile Borel and Francesco Paolo Cantelli, who stated it in the first decades of the 20th century.
Statement of lemma for probability spaces Let E1, E2, ... 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. That is, lim sup En is the outcome that infinitely many of the infinite sequence of events (En) actually occur. Explicitly,
lim sup n → ∞ E n = ⋂ n = 1 ∞ ⋃ k = n ∞ E k . {\displaystyle \limsup _{n\to \infty }E_{n}=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }E_{k}.} The set lim sup En is sometimes denoted {En i.o.}, where "i.o." stands for "infinitely often". The theorem therefore asserts that if the sum of the probabilities of the events En is finite, then the set of all outcomes that contain infinitely many events must have probability zero. Note that no assumption of independence is required.
Example 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 ΣPr(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 Let (En) be a sequence of events in some probability space. The sequence of events { ⋃ n = N ∞ E n } N = 1 ∞ {\textstyle \left\{\bigcup _{n=N}^{\infty }E_{n}\right\}_{N=1}^{\infty }} is non-increasing:
⋃ n = 1 ∞ E n ⊇ ⋃ n = 2 ∞ E n ⊇ ⋯ ⊇ ⋃ n = N ∞ E n ⊇ ⋃ n = N + 1 ∞ E n ⊇ ⋯ ⊇ lim sup n → ∞ E n . {\displaystyle \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 ( lim sup n → ∞ E n ) = lim N → ∞ Pr ( ⋃ n = N ∞ E n ) . {\displaystyle \Pr(\limsup _{n\to \infty }E_{n})=\lim _{N\to \infty }\Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right).} By subadditivity,
Pr ( ⋃ n = N ∞ E n ) ≤ ∑ n = N ∞ Pr ( E n ) . {\displaystyle \Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right)\leq \sum _{n=N}^{\infty }\Pr(E_{n}).} By original assumption, ∑ n = 1 ∞ Pr ( E n ) < ∞ . {\textstyle \sum _{n=1}^{\infty }\Pr(E_{n})<\infty .} As the series ∑ n = 1 ∞ Pr ( E n ) {\textstyle \sum _{n=1}^{\infty }\Pr(E_{n})} converges,
lim N → ∞ ∑ n = N ∞ Pr ( E n ) = 0 , {\displaystyle \lim _{N\to \infty }\sum _{n=N}^{\infty }\Pr(E_{n})=0,}
as required.
General measure spaces For general measure spaces, 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 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:
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 this second lemma.
Example 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
∑ j μ ( E j ) = ∞ , {\displaystyle \sum _{j}\mu (E_{j})=\infty ,}
then there is a sequence Fj of translates
F j = E j + x j {\displaystyle F_{j}=E_{j}+x_{j}}
such that
lim sup j → ∞ F j = ⋂ n = 1 ∞ ⋃ k = n ∞ F k = R n {\displaystyle \limsup _{j\to \infty }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 ∑ n = 1 ∞ Pr ( E n ) = ∞ {\textstyle \sum _{n=1}^{\infty }\Pr(E_{n})=\infty } and the events ( E n ) n = 1 ∞ {\displaystyle (E_{n})_{n=1}^{\infty }} 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
1 − Pr ( lim sup n → ∞ E n ) = 0. {\displaystyle 1-\Pr(\limsup _{n\to \infty }E_{n})=0.}
Noting that:
1 − Pr ( lim sup n → ∞ E n ) = 1 − Pr ( { E n i.o. } ) = Pr ( { E n i.o. } c ) = Pr ( ( ⋂ N = 1 ∞ ⋃ n = N ∞ E n ) c ) = Pr ( ⋃ N = 1 ∞ ⋂ n = N ∞ E n c ) = Pr ( lim inf n → ∞ E n c ) = lim N → ∞ Pr ( ⋂ n = N ∞ E n c ) , {\displaystyle {\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: Pr ( ⋂ n = N ∞ E n c ) = 0 {\textstyle \Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right)=0} . Since the ( E n ) n = 1 ∞ {\displaystyle (E_{n})_{n=1}^{\infty }} are independent:
Pr ( ⋂ n = N ∞ E n c ) = ∏ n = N ∞ Pr ( E n c ) = ∏ n = N ∞ ( 1 − Pr ( E n ) ) . {\displaystyle {\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 ∑ n = N ∞ Pr ( E n ) {\textstyle \sum _{n=N}^{\infty }\Pr(E_{n})} diverges. This completes the proof.
Generalizations
Renyi–Lamperti lemma The assumption of independence in the second lemma can be relaxed. The Renyi–Lamperti lemma states that if the events ( A n ) {\displaystyle (A_{n})} satisfy ∑ Pr ( A n ) = ∞ {\displaystyle \sum \Pr(A_{n})=\infty } and a condition of weak dependence regarding the correlation of the events, specifically:
lim inf n → ∞ ∑ 1 ≤ i , j ≤ n Pr ( A i ∩ A j ) ( ∑ i = 1 n Pr ( A i ) ) 2 = 1 , {\displaystyle \liminf _{n\to \infty }{\frac {\sum _{1\leq i,j\leq n}\Pr(A_{i}\cap A_{j})}{\left(\sum _{i=1}^{n}\Pr(A_{i})\right)^{2}}}=1,}
then Pr ( A n i.o. ) = 1 {\displaystyle \Pr(A_{n}{\text{ i.o.}})=1} . This result is related to the Kochen–Stone theorem, which provides a lower bound for the probability of infinitely many events occurring when the limit inferior in the condition above is positive but not necessarily 1.
Conditional Borel–Cantelli lemma A powerful generalization involving conditional probability is known as the Conditional Borel–Cantelli lemma (or Lévy's extension of the Borel–Cantelli lemma). It connects the occurrence of events to the accumulation of their conditional probabilities given the past. Let ( F n ) {\displaystyle ({\mathcal {F}}_{n})} be a filtration on a probability space, and let E n ∈ F n {\displaystyle E_{n}\in {\mathcal {F}}_{n}} be a sequence of events adapted to the filtration. Then, almost surely:
{ ∑ n = 1 ∞ Pr ( E n ∣ F n − 1 ) = ∞ } = { E n i.o. } . {\displaystyle \left\{\sum _{n=1}^{\infty }\Pr(E_{n}\mid {\mathcal {F}}_{n-1})=\infty \right\}=\{E_{n}{\text{ i.o.}}\}.}
In other words, the event that E n {\displaystyle E_{n}} occurs infinitely often is almost surely equivalent to the event that the sum of the conditional probabilities diverges. This result is a consequence of martingale convergence theorems.
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 ) {\displaystyle (A_{n})} is monotone increasing for sufficiently large indices. This Lemma says: Let ( A n ) {\displaystyle (A_{n})} be such that A k ⊆ A k + 1 {\displaystyle A_{k}\subseteq A_{k+1}} , and let A ¯ {\displaystyle {\bar {A}}} denote the complement of A {\displaystyle A} . Then the probability of infinitely many A k {\displaystyle A_{k}} occur (that is, at least one A k {\displaystyle A_{k}} occurs) is one if and only if there exists a strictly increasing sequence of positive integers ( t k ) {\displaystyle (t_{k})} such that
∑ k Pr ( A t k + 1 ∣ A ¯ t k ) = ∞ . {\displaystyle \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 ) {\displaystyle (t_{k})} usually being the essence.
Kochen–Stone Let ( A n ) {\displaystyle (A_{n})} be a sequence of events with ∑ Pr ( A n ) = ∞ {\textstyle \sum \Pr(A_{n})=\infty } and
lim sup k → ∞ ( ∑ n = 1 k Pr ( A n ) ) 2 ∑ 1 ≤ m , n ≤ k Pr ( A m ∩ A n ) > 0. {\textstyle \limsup _{k\to \infty }{\frac {\left(\sum _{n=1}^{k}\Pr(A_{n})\right)^{2}}{\sum _{1\leq m,n\leq k}\Pr(A_{m}\cap A_{n})}}>0.} Then there is a positive probability that A n {\displaystyle A_{n}} occur infinitely often.
Proof Let S m , n = ∑ i = m n 1 A i {\displaystyle S_{m,n}=\sum _{i=m}^{n}\mathbf {1} _{A_{i}}} . Then, note that
E [ S m , n ] 2 = ( ∑ i = m n Pr ( A i ) ) 2 {\displaystyle E[S_{m,n}]^{2}=\left(\sum _{i=m}^{n}\Pr(A_{i})\right)^{2}}
and
E [ S m , n 2 ] = ∑ 1 ≤ i ≤ j ≤ n Pr ( A i ∩ A j ) . {\displaystyle E[S_{m,n}^{2}]=\sum _{1\leq i\leq j\leq n}\Pr(A_{i}\cap A_{j}).}
Hence, we know that
lim sup n → ∞ E [ S 1 , n ] 2 E [ S 1 , n 2 ] > 0. {\displaystyle \limsup _{n\to \infty }{\frac {\mathbb {E} [S_{1,n}]^{2}}{\mathbb {E} [S_{1,n}^{2}]}}>0.}
We have that
Pr ( ⋃ i = m n A i ) = Pr ( S m , n > 0 ) . {\displaystyle \Pr \left(\bigcup _{i=m}^{n}A_{i}\right)=\Pr(S_{m,n}>0).}
Now, notice that by the Cauchy-Schwarz Inequality, for any random variable X ≥ 0 {\displaystyle X\geq 0} :
E [ X ] 2 ≤ E [ X 1 { X > 0 } ] 2 ≤ E [ X 2 ] Pr ( X > 0 ) , {\displaystyle \mathbb {E} [X]^{2}\leq \mathbb {E} [X\mathbf {1} _{\{X>0\}}]^{2}\leq \mathbb {E} [X^{2}]\Pr(X>0),}
therefore,
Pr ( S m , n > 0 ) ≥ E [ S m , n ] 2 E [ S m , n 2 ] . {\displaystyle \Pr(S_{m,n}>0)\geq {\frac {\mathbb {E} [S_{m,n}]^{2}}{\mathbb {E} [S_{m,n}^{2}]}}.}
We then have
E [ S m , n ] 2 E [ S m , n 2 ] ≥ E [ S 1 , n − S 1 , m − 1 ] 2 E [ S 1 , n 2 ] . {\displaystyle {\frac {\mathbb {E} [S_{m,n}]^{2}}{\mathbb {E} [S_{m,n}^{2}]}}\geq {\frac {E[S_{1,n}-S_{1,m-1}]^{2}}{E[S_{1,n}^{2}]}}.}
Given m {\displaystyle m} , since lim n → ∞ E [ S 1 , n ] = ∞ {\displaystyle \lim _{n\to \infty }\mathbb {E} [S_{1,n}]=\infty } , we can find n {\displaystyle n} large enough so that
| E [ S 1 , n ] − E [ S 1 , m − 1 ] E [ S 1 , n ] − 1 | < ϵ , {\displaystyle {\biggr |}{\frac {\mathbb {E} [S_{1,n}]-\mathbb {E} [S_{1,m-1}]}{\mathbb {E} [S_{1,n}]}}-1{\biggr |}<\epsilon ,}
for any given ϵ > 0 {\displaystyle \epsilon >0} . Therefore,
lim m → ∞ sup n ≥ m Pr ( ⋃ i = m n A i ) ≥ lim m → ∞ sup n ≥ m E [ S 1 , n ] 2 E [ S 1 , n 2 ] > 0. {\displaystyle \lim _{m\to \infty }\sup _{n\geq m}\Pr \left(\bigcup _{i=m}^{n}A_{i}\right)\geq \l