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

✍ Scribed by Jeffrey J. Hunter

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 766 KB
• Volume: 429
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2007.06.016

✦ Synopsis

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 η i = m j =1 m ij π j as a measure of the "mixing time" or "time to stationarity" in a finite irreducible discrete time Markov chain with stationary distribution {π j } and m ij 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 η i = 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 v i , starting in state i. The v i 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, m

(2) ij , and the variance of the first passage times, in a discrete time Markov chain is carried out leading to some new results.