Phase-Type Distributions and Majorization

Aldous and Shepp recently proved that the Erlang distribution of a given order is the least variable phase-type distribution of that order, in the sense of minimizing the coefficient of variation. Here we prove that it is also least variable in the sense of majorization. We give an example showing that the result does not extend in the obvious way to general distributions with rational transforms and this suggests that the inequality hinges on the Markov property.