Why is Kemeny's constant a constant?

TL;DR

This paper provides a simple physical explanation for Kemeny's constant in finite Markov chains, extending to continuous-time cases and discussing limitations for infinite state spaces, with specific results for birth-and-death processes.

Contribution

It introduces a straightforward physical interpretation of Kemeny's constant that applies to finite and continuous-time Markov chains, and analyzes its limitations in infinite state spaces.

Findings

Physical interpretation extends to continuous-time Markov chains.

Kemeny's constant may be infinite in infinite state spaces.

Interpretation holds for birth-and-death processes with finite constant.

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 {\it 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 constant; discrete-time Markov chains; continuous-time Markov chains; passage times; deviation matrix.