Let $X,Y$ be two independent random variables having the same distribution, centred and with variance 1, $\phi$ is the characteristic function of $X$ and $Y$.
If $X+Y$ and $X-Y$ are independent, show that $\phi(t)=\phi(-t)$ $\forall t$ (consider $\rho(t):=\phi(t)/\phi(-t))$
Then I have to deduce X and Y are gaussian r.v.
Seems quite obvious (on the first point I think I have to show they're both symmetric...how?) but I can't come up with a solution
ps from the previous points of the exercise I know that the characteristic function of X and Y verifies $\phi(2t)=\phi(t)^3\phi(-t)$ and $\phi(t)$ is never equal to $0$