Multivariate Matrix-Exponential Distributions

Mogens Bladt, Bo Friis Nielsen

    Research output: Contribution to journalJournal articleResearchpeer-review


    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.
    Original languageEnglish
    JournalStochastic Models
    Issue number1
    Pages (from-to)1-26
    Publication statusPublished - 2010


    • Continued fractions
    • Hankel matrices
    • Multivariate phase-type distribution
    • Multivariate matrix-exponential


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