Convergence analysis of some multivariate Markov chains using stochastic monotonicity

TL;DR

This paper develops a nonasymptotic convergence analysis for multivariate Markov chains, including models from genetics and ecology, using stochastic monotonicity and partial orderings to derive explicit bounds from arbitrary starting points.

Contribution

It extends convergence bounds to multivariate Markov chains with partial orderings, including nonreversible cases, generalizing previous univariate results.

Findings

Derived explicit nonasymptotic bounds for convergence to stationarity.

Applicable to nonreversible Markov chains and complex multivariate models.

Extended analysis to models not previously considered in the literature.

Abstract

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 Polya 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].