Variances of first passage times in a Markov chain with applications to mixing times

Abstract

In an earlier paper [J.J. Hunter, Mixing times with applications to perturbed Markov chains, Linear Algebra Appl. 417 (2006) 108–123] the author introduced the statistic ηi=∑j=1mmijπj as a measure of the “mixing time” or “time to stationarity” in a finite irreducible discrete time Markov chain with stationary distribution {πj} and mij as the mean first passage time from state i to state j of the Markov chain. This was shown to be independent of the initial state i with ηi=η for all i, minimal in the case of a periodic chain, yet can be arbitrarily large in a variety of situations. In this paper we explore the variance of the mixing time vi, starting in state i. The vi are shown to depend on i and an exploration of recommended starting states, given knowledge of the transition probabilities, is considered. As a preamble, a study of the computation of second moments of the first passage times, mij(2), and the variance of the first passage times, in a discrete time Markov chain is carried out leading to some new results.