Weak law of large numbers and CF
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