ERGODICITY AND STABILITY OF THE CONDITIONAL DISTRIBUTIONS OF NONDEGENERATE MARKOV CHAINS

• 成果类型: Article
• 署名作者: Tong, Xin Thomson; van Handel, Ramon
• 署名单位: Princeton University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP800
• 发表日期: 2012
• 页码: 1495-1540
• 关键词: Asymptotic stability; filter

摘要

We consider a bivariate stationary Markov chain (X-n, Y-n)(n >= 0) in a Polish state space, where only the process (Y-n)(n >= 0) is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process (Pi(n))(n >= 0), where Pi(n) is the conditional distribution of X-n given Y-0, ... ,Y-n. We show that the ergodic and stability properties of (Pi(n))(n >= 0) are inherited from the ergodicity of the unobserved process (X-n)(n >= 0) provided that the Markov chain (X-n, Y-n)(n >= 0) is nondegenerate, that is, its transition kernel is equivalent to the product of independent transition kernels. Our main results generalize, subsume and in some cases correct previous results on the ergodic theory of nonlinear filters.