The Distribution of Mixing Times in Markov Chains

TL;DR

This paper derives the probability distribution and generating functions for the mixing times in discrete irreducible Markov chains, extending previous work on expected and variance of mixing times, and presents new results for recurrence and first passage times in three-state chains.

Contribution

It provides explicit formulas for the distribution of mixing times, extending prior results on moments, and introduces new findings for recurrence and first passage times in three-state Markov chains.

Findings

Derived probability generating functions for mixing times.

Extended previous results on expected and variance of mixing times.

Presented new results for recurrence and first passage times in three-state chains.

Abstract

The distribution of the "mixing time" or the "time to stationarity" in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time starting in state i are derived and special cases explored. This extends the results of the author regarding the expected time to mixing [J.J. Hunter, Mixing times with applications to perturbed Markov chains, Linear Algebra Appl. 417 (2006) 108-123], and the variance of the times to mixing, [J.J. Hunter, Variances of first passage times in a Markov chain with applications to mixing times, Linear Algebra Appl. 429 (2008) 1135-1162]. Some new results for the distribution of recurrence and first passage times in three-state Markov chain are also presented.