Sparse representations of phase-type distributions

A phase-type distribution is the distribution of hitting time in a finite-state Markov chain. A phase-type distribution is triangular if one can find an upper triangular Markovian representation for this distribution. We introduce in this paper an extension of the triangular phase-type distributions, which we call monocyclic distributions. We will show that any phase-type distribution can be represented as a mixture of these simple sparse distributions.