Bilateral matrix-exponential distributions

In this article we define the classes of bilateral and multivariate bilateral matrix-exponential distributions. These distributions have support on the entire real space and have rational moment-generating functions. These distributions extend the class of bilateral phasetype distributions of [1] and the class of multivariate matrix-exponential distributions of [9]. We prove a characterization theorem stating that a random variable has a bilateral multivariate distribution if and only if all linear combinations of the coordinates have a univariate bilateral matrix-exponential distribution. As an application we demonstrate that certain multivariate disions, which are governed by the underlying Markov jump process generating a phasetype distribution, have a bilateral matrix-exponential distribution at the time of absorption, see also [4].