A CLOSURE CHARACTERIZATION OF PHASE-TYPE DISTRIBUTIONS

We characterise the classes of continuous and discrete phase-type dist ributions in the following way. They are known to be closed under conv olutions, mixtures, and the unary 'geometric mixture' operation. We sh ow that the continuous class is the smallest family of distributions t hat is closed under these operations and contains all exponential dist ributions and the point mass at zero. An analogous result holds for th e discrete class. We also show that discrete phase-type distributions can be regarded as R+-rational sequences, in the sense of automata the ory. This allows us to view our characterisation of them as a corollar y of the Kleene-Schutzenberger theorem on the behavior of finite autom ata. We prove moreover that any summable R+-rational sequence is propo rtional to a discrete phase-type distribution.