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Why is Kemeny’s constant a constant?

Part of:Markov processesProbabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  16 January 2019

Dario Bini,
Jeffrey J. Hunter,
Guy Latouche,
Beatrice Meini and
Peter Taylor
Dario Bini*
Affiliation:
University of Pisa
Jeffrey J. Hunter*
Affiliation:
Auckland University of Technology
Guy Latouche*
Affiliation:
Université Libre de Bruxelles
Beatrice Meini*
Affiliation:
University of Pisa
Peter Taylor*
Affiliation:
University of Melbourne
*
* Postal address: Dipartimento di Matematica, University of Pisa, 56127 Pisa, Italy.
*** Postal address: Department of Mathematical Sciences, Auckland University of Technology, 1142 Auckland, New Zealand. Email address: jeffrey.hunter@aut.ac.nz
**** Postal address: Département d'informatique, Université Libre de Bruxelles, 1050 Bruxelles, Belgium. Email address: latouche@ulb.ac.be
* Postal address: Dipartimento di Matematica, University of Pisa, 56127 Pisa, Italy.
****** Postal address: School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia. Email address: taylorpg@unimelb.edu.au
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Abstract

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In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as Kemeny’s constant. Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov chains on a finite state space. For Markov chains with denumerably infinite state space, the constant may be infinite and even if it is finite, there is no guarantee that the physical argument will hold. We show that the physical interpretation does go through for the special case of a birth-and-death process with a finite value of Kemeny’s constant.

Keywords

Kemeny’s constantdiscrete-time Markov chaincontinuous-time Markov chainpassage timedeviation matrix

MSC classification

Primary:60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary:65C40: Computational Markov chains

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018, pp. 1025 - 1036
Copyright
Copyright © Applied Probability Trust 2018 

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