## The Annals of Applied Probability

### Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

#### 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 $Y_{0},\ldots,Y_{n}$. 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.

#### Article information

Source
Ann. Appl. Probab., Volume 22, Number 4 (2012), 1495-1540.

Dates
First available in Project Euclid: 10 August 2012

https://projecteuclid.org/euclid.aoap/1344614202

Digital Object Identifier
doi:10.1214/11-AAP800

Mathematical Reviews number (MathSciNet)
MR2985168

Zentralblatt MATH identifier
1252.60069

#### Citation

Tong, Xin Thomson; van Handel, Ramon. Ergodicity and stability of the conditional distributions of nondegenerate Markov chains. Ann. Appl. Probab. 22 (2012), no. 4, 1495--1540. doi:10.1214/11-AAP800. https://projecteuclid.org/euclid.aoap/1344614202

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