Abstract
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. The distributions extend the class of bilateral phase-type distributions of Ahn and Ramaswami [Stoch. Models 21, 239–259 (2005)] and the class of multivariate matrix-exponential distributions of Bladt and Nielsen [Stoch. Models 26, 1–26 (2010)]. 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 diffusions, which are governed by the underlying Markov jump process generating a phase-type distribution, have a bilateral matrix-exponential distribution at the time of absorption.
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Acknowledgements
Luz Judith Rodriguez Esparza and Bo Friis Nielsen would like to thank the Villum Kann Rasmussen Foundation and the Danish Council for Strategic Research for their support through MTlab a VKR centre of excellence and the UNITE project under Grant 2140-08-0011. Mogens Bladt acknowledges the support from the Mexican Research Council, Conacyt, Grant 48538.
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Appendix
Existence of B + and B −
In what follows, we will give an analysis of the existence of B + and B − assuming that we do not have an atom at zero.
Suppose that the polynomial A(s) can be written as \(A(s) ={ \prod \nolimits }_{j=1}^{r}{(s - {\lambda }_{j})}^{{\nu }_{j}}\) for some r such as ∑ j = 1 rν j = deg(A) and whose poles are given by λ j . Then for
Taylor’s theorem (in the real or complex case) provides a proof of the existence and uniqueness of the partial fraction decomposition and a characterization of the coefficients. If we define
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Bladt, M., Esparza, L.J.R., Nielsen, B.F. (2013). Bilateral Matrix-Exponential Distributions. In: Latouche, G., Ramaswami, V., Sethuraman, J., Sigman, K., Squillante, M., D. Yao, D. (eds) Matrix-Analytic Methods in Stochastic Models. Springer Proceedings in Mathematics & Statistics, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4909-6_3
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DOI: https://doi.org/10.1007/978-1-4614-4909-6_3
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