Separation cut-offs for birth and death chains

• 成果类型: Article
• 署名作者: Diaconis, Persi; Saloff-Coste, Laurent
• 署名单位: Stanford University; Cornell University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000501
• 发表日期: 2006
• 页码: 2098-2122
• 关键词: random-walks; markov-chains; finite-groups; times

摘要

This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cutoff if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.