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Phase-type representations for exponential distributions

Part of: Markov processes

Published online by Cambridge University Press:  04 March 2025

Hansjörg Albrecher ,
Clara Brimnes Gardner Open the ORCID record for Clara Brimnes Gardner [Opens in a new window]  and
Bo Friis Nielsen
Hansjörg Albrecher*
Affiliation:
University of Lausanne
Clara Brimnes Gardner*
Affiliation:
Technical University of Denmark
Bo Friis Nielsen*
Affiliation:
Technical University of Denmark
*
*Postal address: Université de Lausanne, Department of Actuarial Science, Quartier UNIL-Chamberonne, Bâtiment Extranef, CH-1015 Lausanne, Switzerland. Email address: hansjoerg.albrecher@unil.ch
**Postal address: DTU Compute, Richard Petersens Plads, Building 324, 2800 Kgs. Lyngby, Denmark
**Postal address: DTU Compute, Richard Petersens Plads, Building 324, 2800 Kgs. Lyngby, Denmark
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Abstract

This paper characterizes irreducible phase-type representations for exponential distributions. Bean and Green (2000) gave a set of necessary and sufficient conditions for a phase-type distribution with an irreducible generator matrix to be exponential. We extend these conditions to irreducible representations, and we thus give a characterization of all irreducible phase-type representations for exponential distributions. We consider the results in relation to time-reversal of phase-type distributions, PH-simplicity, and the algebraic degree of a phase-type distribution, and we give applications of the results. In particular we give the conditions under which a Coxian distribution becomes exponential, and we construct bivariate exponential distributions. Finally, we translate the main findings to the discrete case of geometric distributions.

Keywords

Markov jump processesgenerator matricesbivariate exponential distributionsphase-type distributions

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 57 , Issue 3 , September 2025 , pp. 969 - 999
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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