First, let me quickly remind you of the two operations: convolution and cross-correlation between 2 function $f$ and $g$, assuming continuous domain.
Cross-Correlation $f \star g$ : $\int f(\tau)g(\tau-t)d\tau$
Convolution: $f \otimes g$: $\int f(\tau)g(t-\tau)d\tau$
The difference being in convolution, besides shifting one function, say $g$, like cross-correlation, you reverse it first. I have read works which take convolution of 2 distributions. My question is, why would you take convolution when you can take the cross-correlation? It looks to me as though cross-correlation of probability functions has very obvious interpretative sense while convolution after the reverse of one function, there is no particular interpretive meaning.