Multivariate matrix-exponential distributions

In this extended abstract we define a class of distributions which we shall refer to as multivariate matrix–exponential distributions (MVME). They are defined in a natural way, inspired by the definition of univariate matrix– exponential distributions, as the distributions on R+ having a rational (multidimensional) Laplace transform. A multidimensional rational function is the fraction between two multidimensional polynomials. The marginal distributions are hence univariate matrix–exponential distributions and in general dependent. In one dimension, matrix–exponential distributions are defined as distributions on R+ with a rational Laplace transform, which in turn is equivalent to its density being a weighted sum of the elements of a matrix– exponential. Thereby their name. The main purpose of this work is to characterize the MVME distributions in terms of one–dimensional matrix–exponential distributions. In Section 2 we provide some background on univariate matrix–exponential distributions as well as a historical review of two subclasses of MVME which were defined previously by respectively Assaf et al. and Kulkarni. We also formulate Kulkarni’s definition in terms of the structure of their projections which is more in line with our characterization theorem. The main result states that a multivariate distribution is a MVME if and only if any non–negative non– null linear combination of the coordinates are again matrix–exponential. This