Borel–Cantelli lemma

The Borel - Cantelli lemma (after Émile Borel and Francesco Cantelli ) is a set of probability theory. It is often helpful in the study of almost sure convergence of random variables and is therefore used for the proof of the strong law of large numbers. Another illustrative application of the lemma is the Infinite Monkey Theorem. The lemma consists of two parts, the "classic" set of Borel - Cantelli only contains the first part. The second is an extension and comes from Paul Erdős and Rényi Alfréd.

Statement of the lemma

Formulation

It is an infinite sequence of random events. Then says the Borel - Cantelli lemma

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

Formal statement

Symbolically: For

Applies:

For the proof

The classic statement 1 can be proved: the probability that some event occurs with, is not greater than and strives for the assumed convergence of the sum to 0 for. The limes superior is the event that infinitely many occur, and is a partial event of any of the events mentioned in the previous sentence, and its probability is thus not greater than all members of a null sequence, ie 0, which was to be proved.

Application

From the lemma of Borel - Cantelli results in the following useful criterion for the almost sure convergence of random variables:

Let be a random variable and a sequence of random variables on some probability space.

If for each, then applies almost certain.