  In this paper, we analyze the set of all possible aggregate distributions of
the sum of standard uniform random variables, a simply stated yet challenging
problem in the literature of distributions with given margins. Our main results
are obtained for two distinct cases. In the case of dimension two, we obtain
four partial characterization results. An analytical result with full
generality is not available in this case, and it seems to be out of reach with
existing techniques. For dimension greater or equal to three, we obtain a full
characterization of the set of aggregate distributions, which is the first
complete characterization result of this type in the literature for any choice
of continuous marginal distributions.
