Multivariate Matrix-Exponential Distributions

ISSN： 1532-6349

Source： Stochastic Models, Vol.26, Iss.1, 2010-01, pp. : 1-26

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Abstract

In this article we consider the distributions of non-negative random vectors with a joint rational Laplace transform, i.e., a fraction between two multi-dimensional polynomials. These distributions are in the univariate case known as matrix-exponential distributions, since their densities can be written as linear combinations of the elements in the exponential of a matrix. For this reason we shall refer to multivariate distributions with rational Laplace transform as multivariate matrix-exponential distributions (MVME). The marginal distributions of an MVME are univariate matrix-exponential distributions. We prove a characterization that states that a distribution is an MVME distribution if and only if all non-negative, non-null linear combinations of the coordinates have a univariate matrix-exponential distribution. This theorem is analog to a well-known characterization theorem for the multivariate normal distribution. However, the proof is different and involves theory for rational function based on continued fractions and Hankel determinants.