[Math] Prove that Cauchy distribution is stable

Question

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.

Answer 1

The characteristic function for a Cauchy r.v centered around zero and with scale $\gamma$ is $\exp(- \gamma |t|)$.

If $X_i$ are r.v. with characteristic function $\psi_i(t)$, then $aX_i$ has characteristic function $\psi_i(at)$ and characteristic function of $\sum X_i$ where $X_i$ are independent is $\prod \psi_i(t)$.

Use the above information to conclude what you want.