Mixing times with applications to perturbed Markov chains

A measure of the "mixing time" or "time to stationarity" in a finite irreducible discrete time Markov chain is considered. The statistic eta(i) = Sigma(m)(j=1) m(ij) pi(j), where m(ij) I is the stationary distribution and m(ij) is the mean first passage time from state i to state j of the Markov chain, is shown to be independent of the initial state i (so that eta(i) = eta for all i), is minimal in the case of a periodic chain, yet can be arbitrarily large in a variety of situations. An application considering the effects perturbations of the transition probabilities have on the stationary distributions of Markov chains leads to a new bound, involving eta, for the l-norm of the difference between the stationary probability vectors of the original and the perturbed chain. When eta is large the stationary distribution of the Markov chain is very sensitive to perturbations of the transition probabilities. (c) 2006 Elsevier Inc. All rights reserved.