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Recall that the sequence (Yn ) nN of random variables defined on the probability space
(, , P) converges near-certainly towards c if and only if
P(  , (Yn ( )) nN converges towards c) = 1.
The purpose of this exercise is to prove the following result:
Strong law of large numbers:
Let ( X n ) nN * 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 ) nN 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 , PYn  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.