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Recall that the sequence (Yn ) nN of random variables defined on the probability space (, , P) converges near-certainly towards c if and only if P( , (Yn ( )) nN converges towards c) = 1. The purpose of this exercise is to prove the following result: Strong law of large numbers: Let ( X n ) nN * be a sequence of independent random variables with identical laws, such that E ( X 14 ) , defined on the probability space (, , P) . We denote: n 1 S n X i and Yn S n . n i 1 With these assumptions, the theorem states that (Yn ) nN converges near-certainly towards E( X 1 ) . 1. Prove that E (( X 1 E ( X 1 )) 4 ) , which we denote 4 , is finite, that E S n nE ( X 1 ) 4 is finite, and that E ( X 1 ) , which we denote m, is finite. Prove that for all 0 , PYn p E S n nE X 1 n 4 4 4 . 2. Among the following terms, identify those which are equal to 0, and majorate the others as a function of 4 (i.e., find a number, expressed in terms of 4 , which is greater than or equal to the term): E (( X 1 m) 3 ( X 2 m)), E (( X 1 m) 2 ( X 2 m) 2 ), E (( X 1 m) 2 ( X 2 m)( X 3 m)), E (( X 1 m)( X 2 m)( X 3 m)( X 4 m)). 3. Deduce that there exists a constant C such that for all n N * : C P ( Yn p ) 4 2 . n 4. Denote n 1 1 5 and An {Yn p n } . Using Borel-Cantelli’s lemma, prove that n P(lim sup An) = 0 and conclude.