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 $\displaystyle X_1,X_2,... $ r.v. i.i.d. , with a common mean and variance $\displaystyle \mu $ and $\displaystyle \sigma^2$.

The first think i would like to prove is that

$\displaystyle Var \left ( \frac{1}{n} \sum_{1}^{n} X_i \right ) \underset{n\rightarrow \infty}{\longrightarrow } 0$

So i define $\displaystyle Z_n= \sum_{0}^{n}\frac{X_n}{n}$ as the partial sum. Its characteristic functions is (they are all independent):

$\displaystyle \varphi_{Z_n}(t) =\varphi_{\sum_{0}^{n}\frac{X_n}{n}}(t)=$

$\displaystyle \prod_{0}^{n} \varphi_{\frac{X_i}{n}} = \left [ \varphi_{X/n} \right ]^n $

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:

$\displaystyle \varphi^{(2)}_{Z_n}(0)=\frac{n-1}{n} \mu^2 + \frac{\sigma^2}{n^2}$ which obviously does not goes to 0 (unless $\displaystyle \mu=0$ 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)$\displaystyle Z_n \overset{\mathcal{L}_2}{\longrightarrow}\mu $ (in L2 norm)

3)$\displaystyle Z_n \overset{P}{\longrightarrow}\mu $ (in probability)

Any comments will be much apreciated. thank you