A geometric invariant in weak lumpability of finite Markov chains

A geometric invariant in weak lumpability of finite Markov chains

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Article ID: iaor1999345
Country: United Kingdom
Volume: 34
Issue: 4
Start Page Number: 847
End Page Number: 858
Publication Date: Dec 1997
Journal: Journal of Applied Probability
Authors:
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.

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