Borel-Cantelli Lemma

The Borel-Cantelli lemma , sometimes also Borel’s zero-one law , (after Émile Borel and Francesco Cantelli ) is a set of probability theory . It is often helpful in examining the almost certain convergence of random variables and is therefore used to prove the strong law of large numbers . Another illustrative application of the lemma is the Infinite Monkey Theorem . The lemma consists of two parts, with Borel-Cantelli's "classical" theorem only containing the first part. The second is an extension and comes from Paul Erdős and Alfréd Rényi .

Statement of the lemma

formulation

The Borel-Cantelli lemma says:

Let it be an infinite sequence of events in a probability space . ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$

Then:

1. If the sum of the probabilities of the finite, then the probability of Limes superior of equal 0th ${\displaystyle A_{n}}$ ${\displaystyle A_{n}}$
2. If the sum of the probabilities of the is infinite and the events are at least pairwise independent, then the probability of the limit superior is equal to 1. ${\displaystyle A_{n}}$ ${\displaystyle A_{n}}$ ${\displaystyle A_{n}}$

Since the statement is of the form that the probability of a set, here the limes superior, is either 0 or 1, the Borel-Cantelli lemma is one of the 0-1 laws .

Formal statement

Symbolic: for

${\displaystyle A=\limsup _{n\rightarrow \infty }{A_{n}}=\bigcap _{n=1}^{\infty }\bigcup _{i=n}^{\infty }A_{i}=\{A_{n}{\rm {{\,\,unendlich\,\,oft}\}}}}$

${\displaystyle =\{\omega \in \Omega :\omega \in A_{n}{\rm {{\,\,f{\ddot {u}}r\,\,unendlich\,\,viele\,\,}n\in \mathbb {N} \}}}}$

applies:

1. ${\displaystyle \sum _{n\geq 1}P(A_{n})<\infty \Rightarrow P(A)=0}$
2. ${\displaystyle \sum _{n\geq 1}P(A_{n})=\infty }$and they are independent in pairs ${\displaystyle A_{n}}$ ${\displaystyle \Rightarrow P(A)=1}$

For proof

The classic statement 1 can be proved as follows: The probability of any event with occurs, is not greater than and aims because of the assumed convergence of the sum to 0 for . The limes superior is the event that an infinite number of events occurs and is a partial event of each of the events mentioned in the previous sentence, and its probability is therefore no greater than all terms of a zero sequence, i.e. 0, which was to be proven. ${\displaystyle A_{k}}$ ${\displaystyle k\geq n}$ ${\displaystyle \sum _{k=n}^{\infty }P(A_{k})}$ ${\displaystyle n\to \infty }$ ${\displaystyle A_{n}}$ ${\displaystyle A_{n}}$

application

The following useful criterion for the almost certain convergence of random variables results from Borel-Cantelli's lemma:

Let be a random variable and a sequence of random variables over a certain probability space . ${\displaystyle X}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\Omega ,{\mathcal {F}},P)}$

If for everyone , then almost certainly applies . ${\displaystyle \sum _{n=1}^{\infty }P(|X_{n}-X|>\varepsilon )<\infty }$ ${\displaystyle \varepsilon >0}$ ${\displaystyle X_{n}\rightarrow X}$

Counterpart to the Borel-Cantelli Lemma

A useful “counterpart” to the Borel-Cantelli lemma replaces the pairwise independence of the , which is assumed in the second version, by a monotony hypothesis for all sufficiently large indices k. This lemma says: ${\displaystyle (A_{k})}$

Let be a sequence of events that satisfies k sufficiently large for all, and let the event complement to . Then infinitely many occur with probability 1 if and only if a strictly monotonically increasing sequence exists with ${\displaystyle (A_{k})}$ ${\displaystyle A_{k}\subseteq A_{k+1}}$ ${\displaystyle {\bar {A}}}$ ${\displaystyle A}$ ${\displaystyle A_{k}}$ ${\displaystyle (t_{k})}$

${\displaystyle \sum _{k}P(A_{t_{k+1}}\mid {\bar {A}}_{t_{k}})=\infty .}$

This result is useful for problems that affect probabilities, such as B. the question of whether a stochastic process occurs with probability 1 in a certain set of states. The state set is defined as absorbing, which implies the monotony, and a clever choice of the sequence then often provides the answer quickly. ${\displaystyle (t_{k})}$

swell

• Heinz Bauer : Probability Theory (=  De Gruyter textbook ). 5th, revised and improved edition. de Gruyter , Berlin, New York 2002, ISBN 3-11-017236-4 . MR1902050
• A. Rényi : Probability Theory . With an appendix on information theory (=  university books for mathematics . Volume 54 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1971.
• AN Širjaev : Probability (=  university books for mathematics . Volume 91 ). VEB Deutscher Verlag der Wissenschaften , Berlin 1988, ISBN 3-326-00195-9 . MR0967761
• FT Bruss : A Counterpart of the Borel-Cantelli Lemma (=  Journal of Applied Probability . Volume 17 ). Applied Probability Trust, Sheffield 1980.

Individual evidence

1. ↑ ^{a b} Heinz Bauer: Probability Theory. 2002, p. 73 ff
2. ^ A. Rényi: Probability Theory. 1971, pp. 252, 326 ff
3. ↑ ^{a b} A. N. Širjaev: Probability. 1988, p. 265 ff
4. ↑ : FT Bruss: Counterpart Borel-Cantelli Lemma 1980, p. 1094 ff