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Why is Kemeny’s constant a constant?

  • Dario Bini (a1), Jeffrey J. Hunter (a2), Guy Latouche (a3), Beatrice Meini (a1) and Peter Taylor (a4)...

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.

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Corresponding author

* Postal address: Dipartimento di Matematica, University of Pisa, 56127 Pisa, Italy.
** Email address: bini@dm.unipi.it
*** 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
***** Email address: meini@dm.unipi.it
****** Postal address: School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia. Email address: taylorpg@unimelb.edu.au

References

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