The Distribution of Mixing Times in Markov Chains

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.