Polynomial convergence rates of Markov chains

• 成果类型: Article
• 署名作者: Jarner, SF; Roberts, GO
• 署名单位: Lancaster University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 224-247
• 关键词: stability; hastings

摘要

This paper considers the use of Foster-Liapounov-type drift conditions to establish polynomial rates of convergence of the f-norm for a general state space Markov chain in discrete time. Results of this type involving geometric convergence rates are now well established; see, for example, Meyn and Tweedie (11992) and Chapters 15, 16 of Meyn and Tweedie (1993). However, the more subtle polynomial case is not nearly as well understood. The foundational work of Tuominen and Tweedie (1994) studies general subgeometric rates using a sequence of drift conditions. Our results build on this work, but because we restrict ourselves to polynomial rates we are able to take a different and more direct approach which ultimately leads to the derivation of the single drift condition (1). This condition is the natural analogue of the drift condition for geometric ergodicity and, as will be illustrated by examples, it is simple to apply in practice. Let X = (X-0, X-1,...) be a discrete-time Markov chain on a general state space with transition kernel P. Assume X is psi-irreducible, aperiodic and positive recurrent. A main result of the paper is Theorem 3.6 which states that if there exists a test function V greater than or equal to 1, positive constants c and b, a petite set C and 0 less than or equal to alpha < 1 such that (1) PV less than or equal to V - cV(alpha) + b1(C), then the chain is positive recurrent and there is polynomial convergence of the n-step transition kernel P-n to the invariant distribution pi in the sense that the following statement holds (2) n(beta-1)parallel toP(n)(x,.)-piparallel tov(beta) --> 0, n --> infinity.