I have three independent non-negative random variables $X_1$, $X_2$, and $X_3$, and I do not have their density functions, but I do have a decent upper bound for their cdfs.  In other words, I have functions $G_1$, $G_2$, $G_3$ such that $\Pr(X_i\leq x) \leq G_i(x)$ for all $i$ and $x$.  Is there any way at all that I could use these functions to bound the cdf of the sum, $\Pr(X_1+X_2+X_3\leq x)$?  This seems hopeless but I figured I'd give it a shot.