We review what is currently known about one-dimensional distributions on the non-negative reals with rational Laplace transform, also known as matrix-exponential distributions. In particular we discuss a flow interpreation which enables one to mimic certain probabilisticly inspired arguments which are known from the theory of phase-type distributions. We then move on to present ongoing research for higher dimensions. We discuss a characterization result, some closure properties, and a number of examples. Finally we present open problems and future perspectives.
@InProceedings{bladt_et_al:DagSemProc.07461.10, author = {Bladt, Mogens and Nielsen, Bo Friis}, title = {{Multivariate matrix-exponential distributions}}, booktitle = {Numerical Methods for Structured Markov Chains}, pages = {1--13}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7461}, editor = {Dario Bini and Beatrice Meini and Vaidyanathan Ramaswami and Marie-Ange Remiche and Peter Taylor}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07461.10}, URN = {urn:nbn:de:0030-drops-13975}, doi = {10.4230/DagSemProc.07461.10}, annote = {Keywords: Multivariate matrix-exponential distributions, multivariate phase-type distributions, rational Laplace transform} }
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