# Limit of characteristic functions

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?

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Do you know the central limit theorem? – Nate Eldredge Jun 14 '13 at 13:20

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.