I'm trying to prove that the Cauchy distribution is stable, i.e., if $X_{1}, X_{2}, ...$ are i.i.d. Cauchy random variables then $\frac{1}{n}(X_{1}+...+X_{n})$ has the same distribution as $X_{1}$ for $n \geq 1$.
I suspect the proof has something to do with characteristic functions, but haven't been able to write it out. Anyone have any hints on how to approach this?
Thanks.