Let $X,Y$ be two independent (and identically distributed) random variables. Let $Z:=X+Y$.
It's easy to check that the moment generating function $\phi_Z(t):=\mathbb{E}[\,e^{itZ}\,]$ can be expressed as $\phi_Z=\phi_X\cdot\phi_Y$.
Is there a way to express the cumulative distribution function $F_Z(z):=\mathbb{P}(Z\leq z)$ using the cumulative distribution functions of $X$ and $Y$ ?
Edit: note I don't assume that $X$ and $Y$ have a density.
Best Answer
\begin{eqnarray*} F_Z \left( z \right) & = & \int F_X \left( z - y \right) dF_Y \left( y \right)\\ & = & \int F_Y \left( z - x \right) dF_X \left( x \right) \end{eqnarray*}