Please see the attached file for full problem description.
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:
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.
Questions regarding probability spaces are answered. The solution is detailed and well presented.