I have been searching but have thus far been unable to find whether there is a formula for convolving a distribution function $n$ times with itself. For example, given $g:\mathbb{R}\to \mathbb{R}$ is integrable and has compact support, is there a relatively neat formula for $g*\overset{n-\text{times}}{....}g$? For example if $g(x)=x^d\chi_{[0,1)}(x)$ where $d\in\mathbb{N}$?
I know about some particular functions who gave known results, mostly arising from probability, but I was wondering whether there is some more general result shich I couldn't find?