Ergodicity and stability of the conditional distributions of nondegenerate markov chains

Xin Thomson Tong, Ramon Van Handel

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

We consider a bivariate stationary Markov chain (Xn,Y n)n≥0 in a Polish state space, where only the process (Yn)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 (σn)n≥0, where σn is the conditional distribution of Xn given Y0,.., Yn. We show that the ergodic and stability properties of (σn)n≥0 are inherited from the ergodicity of the unobserved process (Xn)n≥0 provided that the Markov chain (Xn,Yn)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.

Original languageEnglish (US)
Pages (from-to)1495-1540
Number of pages46
JournalAnnals of Applied Probability
Volume22
Issue number4
DOIs
StatePublished - Aug 1 2012

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Asymptotic stability
  • Exchange of intersection and supremum
  • Markov chain in random environment
  • Nondegenerate Markov chains
  • Nonlinear filtering
  • Unique ergodicity

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