Proof by characteristic functions that $X+Y$ and $2X$ are identically distributed

Question

The exercise states that X,Y iid and we know that X+Y has Cauchy distribution. And they require to prove that 2X has also Cauchy distribution. Let me put it straight, I dont think I understand it fully , let's forget about Cauchy distribution at all.

Isn't it true for all distributions? If two random variables are iid, their characteristic functions are equal and

$$Ee^{itX}=Ee^{itY}$$ $$Ee^{it(X+Y)}=Ee^{itX}Ee^{itY}=Ee^{itX}Ee^{itX}=Ee^{it2X}$$ hence $2X$ has distribution as $X+Y$ almost surely?

Answer 1

The characteristic function of $X$ is $\phi(t):=\exp (ix_0t-\gamma|t|)$. The desired result is $\phi^2(t)=\phi(2t)$.

Answer 2

As pointed out by Lord Shark the Unknown, the error in your general proof is $(E e^{itX})^2 = E e^{it2X}$, which does not hold in general.

However, as you noted, in the case of $X$ being Cauchy with scale $\gamma$, the above becomes $$(e^{-\gamma|t|})^2 = e^{-\gamma |2t|}.$$