Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

TL;DR

This paper studies the ergodic and stability properties of the conditional distributions in a bivariate Markov chain, showing they inherit properties from the unobserved process under nondegeneracy conditions, extending previous results.

Contribution

It generalizes and corrects prior work on the ergodic theory of nonlinear filters for nondegenerate Markov chains.

Findings

Conditional distributions inherit ergodic properties from the unobserved process.

Nondegeneracy of the Markov chain ensures stability of the conditional distributions.

Results extend and refine previous ergodic theory results for nonlinear filters.

Abstract

We consider a bivariate stationary Markov chain $(X_n,Y_n)_{n\ge0}$ in a Polish state space, where only the process $(Y_n)_{n\ge0}$ is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process $(\Pi_n)_{n\ge0}$, where $\Pi_n$ is the conditional distribution of $X_n$ given $Y_0,...,Y_n$. We show that the ergodic and stability properties of $(\Pi_n)_{n\ge0}$ are inherited from the ergodicity of the unobserved process $(X_n)_{n\ge0}$ provided that the Markov chain $(X_n,Y_n)_{n\ge0}$ 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.