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A geometric invariant in weak lumpability of finite Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

James Ledoux
James Ledoux*
Affiliation:
INSA
*
∗ Postal address: INSA, 20 Avenue des Buttes de Cöesmes 35043 Rennes Cedex, France.
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Abstract

We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.

Keywords

STATES AGGREGATIONPOSITIVE INVARIANCE OF CONES

MSC classification

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

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 847 - 858
Copyright
Copyright © Applied Probability Trust 1997 

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