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Lumpability and marginalisability for continuous-time Markov chains

Part of: Markov processes Physiological, cellular and medical topics

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Geoffrey F. Yeo
Frank Ball*
Affiliation:
University of Nottingham
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗ Postal address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150, Australia.
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Abstract

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

Keywords

STRONG AND WEAK LUMPABILITYTIME REVERSIBLE PROCESSION CHANNEL MODELLINGBIRTH–DEATH PROCESSES

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92C05: Biophysics 92C30: Physiology (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 518 - 528
Copyright
Copyright © Applied Probability Trust 1993 

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