if $\phi$ is a characteristic function, then $|\phi|^2$ is also a characteristic function

Question

Show that if $\phi$ is a characteristic function of some random variable, then $|\phi|^2$ is also a characteristic function.

Answer 1

If $X$ and $Y$ are independent random variables with characteristic function $\phi$ the $Ee^{it(X-Y)}=|Ee^{itX}|^{2}$ so the answer is YES.