CONVERGENCE ANALYSIS OF SOME MULTIVARIATE MARKOV CHAINS USING STOCHASTIC MONOTONICITY

• 成果类型: Article
• 署名作者: Khare, Kshitij; Mukherjee, Nabanita
• 署名单位: State University System of Florida; University of Florida; University of Pennsylvania; Pennsylvania Medicine; Childrens Hospital of Philadelphia
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP856
• 发表日期: 2013
• 页码: 811-833
• 关键词: rates

摘要

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].