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A decomposition theorem for infinite stochastic matrices

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Daniel P. Heyman
Daniel P. Heyman*
Affiliation:
Bellcore
*
∗Postal address: Bellcore, Room 3D-308, 331 Newman Springs Road, Red Bank, NJ 0771–7020, USA.
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Abstract

We prove that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I – P=(A – I)(B – S), where A is strictly upper-triangular, B is strictly lower-triangular, and S is diagonal. Moreover, the elements of A are expected values of random variables that we will specify, and the elements of B and S are probabilities of events that we will specify. The decomposition can be used to obtain steady-state probabilities, mean first-passage-times and the fundamental matrix.

Keywords

MARKOV CHAINSLU FACTORIZATIONSTEADY-STATE PROBABILITIESFIRST-PASSAGE-TIMESFUNDAMENTAL MATRIX

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 893 - 901
Copyright
Copyright © Applied Probability Trust 1995 

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