The Annals of Applied Probability

Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

Xin Thomson Tong and Ramon van Handel

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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.

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

First available in Project Euclid: 10 August 2012

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Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 28D99: None of the above, but in this section
Secondary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 93E11: Filtering [See also 60G35] 93E15: Stochastic stability

Nonlinear filtering unique ergodicity asymptotic stability nondegenerate Markov chains exchange of intersection and supremum Markov chain in random environment


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.

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