On the Limitations of Multivariate Phase-Type Families

Consider a finite set of independent phase-type random variables. Suppose that we construct a random vector, each component of which is a total over a subset of this set. In 1984, D. Assaf et al. conjectured that such a vector need not have a multivariate phase-type distribution, and they provided a sufficient condition for the distribution to be of that type. We confirm their conjecture, and show that their sufficient condition is also essentially necessary. An open question of V. G. Kulkarni is whether or not his extension of the multivariate phase-type family is closed under familiar operations arising in reliability. This question is answered here in the negative.