Why is Kemeny's constant a constant?. (16th January 2019)

• Record Type: Journal Article
• Authors: Bini, Dario; Hunter, Jeffrey J.; Latouche, Guy; Meini, Beatrice; Taylor, Peter
• Is Part Of: Journal of applied probability. Volume 55:Number 4(2018)
• Page Start: 1025
• Page End: 1036
• Publication Date: 2019-01-16
• Subjects: Kemeny's constant, -- discrete-time Markov chain, -- continuous-time Markov chain, -- passage time, -- deviation matrix; Primary 60J10, -- Secondary 65C40; 519.2
• DOI: 10.1017/jpr.2018.68
• Languages: English
• ISSNs: 0021-9002

Abstract

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.