Let $\xi_1 ... , \xi_n$ be iid with $E \xi_i^2 < \infty $ what is $$ \lim _{n\rightarrow \infty} \varphi_{\bar{\xi}}$$ where $\varphi$ is the caracteristic function and $\bar{\xi}$ the mean of all the $\xi_i$.
I don't know how to solve this, can someone help me please.
solutioin (?): $\varphi_{\bar{\xi(t)}} = \varphi_{\xi(t/n)}^n = \left[ 1 + i \mu \frac{t}{n} + O(\frac{t}{n})\right]^n = e^{it\mu}$. By taylor expansion where $\mu$ is the mean of $\xi_i$. Have I missed something?
No, you didn't miss anything. What you proved is that $\bar\xi_n$ converges in law to the constant $\mu$. We can deduce from that that there is convergence in probability to this constant, which is nothing but weak law of large numbers.