Borel-Cantelli Lemma

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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


The Borel-Cantelli lemma says:

Let it be an infinite sequence of events in a probability space .


  1. If the sum of the probabilities of the finite, then the probability of Limes superior of equal 0th
  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.

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


  1. and they are independent in pairs

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.


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 .

If for everyone , then almost certainly applies .

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:

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

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.


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