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Why is Kemeny’s constant a constant?

Part of: Markov processes Probabilistic 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: [email protected]
**** Postal address: Département d'informatique, Université Libre de Bruxelles, 1050 Bruxelles, Belgium. Email address: [email protected]
* 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: [email protected]
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Abstract

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
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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References

[1]Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.Google Scholar
[2]Bansaye, V., Méléard, S. and Richard, M. (2016). Speed of coming down from infinity for birth and death processes. Adv. Appl. Prob. 48, 1183–1210.Google Scholar
[3]Campbell, S. L. and MeyerC. D., Jr. C. D., Jr. (1991). Generalised Inverses of Linear Transformations. Dover Publications, New York.Google Scholar
[4]Coolen-Schrijner, P. and van Doorn, E. A. (2002). The deviation matrix of a continuous-time Markov chain. Prob. Eng. Informat. Sci. 16, 351–366.Google Scholar
[5]Doyle, P. G. (2009). The Kemeny constant of a Markov chain. Preprint. Available at http://arxiv.org/abs/0909.2636.Google Scholar
[6]Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pures Appl. 38, 301–345.Google Scholar
[7]Grinstead, C. M. and Snell, J. L. (1997). Introduction to Probability. AMS, Providence, RI.Google Scholar
[8]Hunter, J. J. (2006). Mixing times with applications to perturbed Markov chains. Linear Algebra Appl. 417, 108–123.Google Scholar
[9]Hunter, J. J. (2014). The role of Kemeny’s constant in properties of Markov chains. Commun. Statist. Theory Meth. 43, 1309–1321.Google Scholar
[10]Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, Princeton, NJ..Google Scholar
[11]Kijima, M. (1997). Markov Processes for Stochastic Modelling. Chapman and Hall, London.Google Scholar
[12]Mao, Y. (2002). Strong ergodicity for Markov processes by coupling methods. J. Appl. Prob. 39, 839–852.Google Scholar
[13]Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Cambridge, MA.Google Scholar
[14]Seneta, E. (1981). Non-negative Matrices and Markov chains, 2nd edn. Springer, New York.Google Scholar
[15]Syski, R. (1978). Ergodic potential. Stoch. Process. Appl. 7, 311–336.Google Scholar
[16]Van Doorn, E. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.Google Scholar
[17]Zhang, H., Chen, A., Lin, X. and Zhang, Y. (2001). Strong ergodicity of monotone transition functions. Statist. Prob. Lett. 55, 63–69.Google Scholar
[18]Zhang, Y. (2001). Strong ergodicity for single-birth processes. J. Appl. Prob. 38, 270–277.Google Scholar