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April 2013 Convergence analysis of some multivariate Markov chains using stochastic monotonicity
Kshitij Khare, Nabanita Mukherjee
Ann. Appl. Probab. 23(2): 811-833 (April 2013). DOI: 10.1214/12-AAP856


We provide a nonasymptotic analysis of convergence to stationarity for a collection of Markov chains on multivariate state spaces, from arbitrary starting points, thereby generalizing results in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737–777]. Our examples include the multi-allele Moran model in population genetics and its variants in community ecology, a generalized Ehrenfest urn model and variants of the Pólya urn model. It is shown that all these Markov chains are stochastically monotone with respect to an appropriate partial ordering. Then, using a generalization of the results in [Diaconis, Khare and Saloff-Coste Sankhya 72 (2010) 45–76] and [Wilson Ann. Appl. Probab. 14 (2004) 274–325] (for univariate totally ordered spaces) to multivariate partially ordered spaces, we obtain explicit nonasymptotic bounds for the distance to stationarity from arbitrary starting points. In previous literature, bounds, if any, were available only from special starting points. The analysis also works for nonreversible Markov chains, and allows us to analyze cases of the multi-allele Moran model not considered in [Khare and Zhou Ann. Appl. Probab. 19 (2009) 737–777].


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Kshitij Khare. Nabanita Mukherjee. "Convergence analysis of some multivariate Markov chains using stochastic monotonicity." Ann. Appl. Probab. 23 (2) 811 - 833, April 2013.


Published: April 2013
First available in Project Euclid: 12 February 2013

zbMATH: 1277.60117
MathSciNet: MR3059276
Digital Object Identifier: 10.1214/12-AAP856

Primary: 60J10
Secondary: 47H05 , 60J22

Keywords: Convergence analysis , Markov chains , partial ordering , Stochastic monotonicity

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 2 • April 2013
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