I am asked to prove the weak law of large numbers using characteristic functions. For simplicity i can assume that the r.v. are simmetryc (so their CF is real).

Let r.v. i.i.d. , with a common mean and variance and .

The first think i would like to prove is that

So i define as the partial sum. Its characteristic functions is (they are all independent):

Im interested in the variance (second moment) so i am to diferentiate two times the CF and evaluate at 0. After computing the derivatives carefully using chain rule i get:

which obviously does not goes to 0 (unless which it has not to be true)

So i dont know if i am doing something wrong.

This is only one of the 3 statements that i understand to be "the weak law of large numbers", the other two are as follow:

2) (in L2 norm)

3) (in probability)

Any comments will be much apreciated. thank you