Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

Abstract

We consider a bivariate stationary Markov chain $(X_n,Y_n{)}_{n\ge 0}$ in a Polish state space, where only the process $(Y_n{)}_{n\ge 0}$ is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process $({\Pi }_n{)}_{n\ge 0}$, where ${\Pi }_n$ is the conditional distribution of $X_n$ given $Y_0,...,Y_n$. We show that the ergodic and stability properties of $({\Pi }_n{)}_{n\ge 0}$ are inherited from the ergodicity of the unobserved process $(X_n{)}_{n\ge 0}$ provided that the Markov chain $(X_n,Y_n{)}_{n\ge 0}$ 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.

Published in at http://dx.doi.org/10.1214/11-AAP800 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Uniform Time Average Consistency of Monte Carlo Particle Filters
• The stability of conditional Markov processes and Markov chains in random environments
• On the exchange of intersection and supremum of sigma-fields in filtering theory

Cited by in corpus (2)

• Conditional ergodicity in infinite dimension
• Phase Transitions in Nonlinear Filtering