# 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

${\ displaystyle A = \ limsup _ {n \ rightarrow \ infty} {A_ {n}} = \ bigcap _ {n = 1} ^ {\ infty} \ bigcup _ {i = n} ^ {\ infty} A_ { i} = \ {A_ {n} {\ rm {{\, \, infinite \, \, often} \}}}}$
${\ displaystyle = \ {\ omega \ in \ Omega: \ omega \ in A_ {n} {\ rm {{\, \, f {\ ddot {u}} r \, \, infinite \, \, many \ , \,} n \ in \ mathbb {N} \}}}}$

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. ${\ displaystyle (t_ {k})}$

## 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