Phase-type distributions and invariant polytopes

The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace.Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.