Use the uniqueness of characteristic functions to pove that $\{X_{i}\}_{i\in\mathbb{N}}$ are independent and identically distributed with "Cauchy Law," $S_{n}=X_{1}+\cdots+X_{n}.$ Then $\frac{S_{n}}{n}\rightarrow 0$ in distribution, when $n$ goes to infinity.
I was thinking in use,evidently the uniqueness of characterístic function. So, $\frac{S_{n}}{n}$ has the same distribution, for example, as $X_{1}.$ Then both random variables has the same distribution. But, ¿how can I do to reach to the goal?
I can't see why $\frac{S_{n}}{n}$ convergence in distribution.
Any kind of idea will be thanked in advance.