# Probabilities : Strong Law of Large Numbers

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Recall that the sequence of random variables defined on the probability space converges near-certainly towards c if and only if

converges towards c) = 1.

The purpose of this exercise is to prove the following result:

Strong law of large numbers:

Let be a sequence of independent random variables with identical laws, such that , defined on the probability space . We denote:

and .

With these assumptions, the theorem states that converges near-certainly towards .

1. Prove that , which we denote , is finite,

that is finite,

and that , which we denote m, is finite.

Prove that for all , .

2. Among the following terms, identify those which are equal to 0, and majorate the others as a function of (i.e., find a number, expressed in terms of , which is greater than or equal to the term):

3. Deduce that there exists a constant C such that for all :

.

4. Denote and . Using Borel-Cantelli's lemma, prove that

P(lim sup An) = 0 and conclude.

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Questions regarding probability spaces are answered. The solution is detailed and well presented.

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