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Lumpability and marginalisability for continuous-time Markov chains

Published online by Cambridge University Press:  14 July 2016

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.

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1993 

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