Showing functions are characteristic

Question

Suppose that $\phi$ is a characteristic function.

How can I prove that $\phi^2$ and $|\phi|^2$ are characteristic functions?

I use the definition of characteristic:

$\phi_x^2(t)=(E(e^{itx}))^2=var(e^{itx})+E(e^{itx})$.

Answer 1

$(E(e^{itX}))^2$ is $E(e^{it(X+X’})$ where $X’$ is an independent copy of $X.$ So the first one is the characteristic function of $X+X’.$

The second one is similarly the characteristic function of $X-X’.$