Characteristic function of the variable X - Y

Question

Let $\phi(X)$ be a characteristic function of $X$. Let $Y$ be independent of $X$ and has the same distribution as $X$. How to find a characteristic function of a variable $X-Y$? I know for a variable $X+Y$ we have: $\phi_{X+Y} (t) =\phi_{X} (t) \phi_{Y} (t)$ but in this case I don't know how to use it..

Answer 1

$X$ and $-Y$ are also independent. So $Ee^{it(X-Y)}=Ee^{it(X+(-Y))}=Ee^{itX}Ee^{-itY}$ and $Ee^{-itY}$ is nothing but the complex conjugate of $\phi_Y(t)$. Hence, $\phi_{X-Y}(t)=\phi_X (t) \overline {\phi_Y(t)}=|\phi_X(t)|^{2}$.