The Kemeny constant of a Markov chain

Given an ergodic finite-state Markov chain, let Miw denote the mean time from i to equilibrium, meaning the expected time, starting from i, to arrive at a state selected randomly according to the equilibrium measure w of the chain. John Kemeny observed that Miw does not depend on starting the point i. The common value K = Miw is the Kemeny constant or seek time of the chain. K is a spectral invariant, to wit, the trace of the resolvent matrix. We review basic facts about the seek time, and connect it to the bus paradox and the Central Limit Theorem for ergodic Markov chains. For J. Laurie Snell The seek time We begin by reviewing basic facts and establishing notation for Markov chains. For background, see Kemeny and Snell [4] or Grinstead and Snell [3], bearing in mind that the notation here is somewhat different. Let P be the transition matrix of an ergodic finite-state Markov chain. We write the entries of P using tensor notation, with P j i being the probability that from state i we move to state j. (There is some possibility ∗Copyright (C) 2009 Peter G. Doyle. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, as published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.