The distribution of Mixing Times in Markov Chains

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(3.16 MB)
Date
2011-11-01
Authors
Hunter, JJ
Supervisor
Item type
Commissioned Report
Degree name
Journal Title
Journal ISSN
Volume Title
Publisher
arXiv
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.

Description
Keywords
Markov chains , Stationary distribution , First passage times , Hitting times , Mixing times , time to stationarity , Kemeny constant , distributions
Source
arXiv:1111.0151
DOI
Publisher's version
http://arxiv.org/abs/1111.0151
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Permanent link
https://hdl.handle.net/10292/3280
Collections
School of Engineering, Computer and Mathematical Sciences - Te Kura Mātai Pūhanga, Rorohiko, Pāngarau