Characterizing phase type distributions

To prove that our moment matching algorithm results in a nearly minimal number of phases, we need to know the minimal number of phases needed to well-represent an input distribution, $G$, by an acyclic PH distribution (namely, OPT($G$)). Thus, we will provide a formal characterization of the set of distributions, ${\cal S}^{(n)}$, that are well-represented by an $n$-phase acyclic PH distribution for each $n=1,2,3,...$.

Definition 5   Let ${\cal S}^{(n)}$ denote the set of distributions that are well-represented by an $n$-phase acyclic PH distribution for positive integer $n$.