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Linear dynamics for the state vector of Markov chain functions

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

James Ledoux
James Ledoux*
Affiliation:
Centre de Mathématiques INSA, Rennes
*
∗ Postal address: Centre de Mathématiques INSA, 20 ave des Buttes de Coesmes, CS 14315, 35043 Rennes cedex, France. Email address: [email protected]
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Abstract

Let (φ(Xn))n be a function of a finite-state Markov chain (Xn)n. In this article, we investigate the conditions under which the random variables φ(n) have the same distribution as Yn (for every n), where (Yn)n is a Markov chain with fixed transition probability matrix. In other words, for a deterministic function φ, we investigate the conditions under which (Xn)n is weakly lumpable for the state vector. We show that the set of all probability distributions of X0, such that (Xn)n is weakly lumpable for the state vector, can be finitely generated. The connections between our definition of lumpability and the usual one (i.e. as the proportional dynamics property) are discussed.

Keywords

Lumpabilityproportional dynamicsRogers-Pitman matrix

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 1198 - 1211
Copyright
Copyright © Applied Probability Trust 2004 

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