Ergodicity and stability of the conditional distributions of nondegenerate markov chains

Xin Thomson Tong; Ramon Van Handel

doi:10.1214/11-AAP800

Ergodicity and stability of the conditional distributions of nondegenerate markov chains

Xin Thomson Tong, Ramon Van Handel

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We consider a bivariate stationary Markov chain (X_n,Y _n)_n≥0 in a Polish state space, where only the process (Y_n)_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 (σ_n)_n≥0, where σ_n is the conditional distribution of X_n given Y₀,.., Y_n. We show that the ergodic and stability properties of (σ_n)_n≥0 are inherited from the ergodicity of the unobserved process (X_n)_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 language	English (US)
Pages (from-to)	1495-1540
Number of pages	46
Journal	Annals of Applied Probability
Volume	22
Issue number	4
DOIs	https://doi.org/10.1214/11-AAP800
State	Published - Aug 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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10.1214/11-AAP800

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title = "Ergodicity and stability of the conditional distributions of nondegenerate markov chains",

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.",

keywords = "Asymptotic stability, Exchange of intersection and supremum, Markov chain in random environment, Nondegenerate Markov chains, Nonlinear filtering, Unique ergodicity",

author = "Tong, {Xin Thomson} and Handel, {Ramon Van}",

year = "2012",

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AU - Handel, Ramon Van

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N2 - 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.

AB - 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.

KW - Asymptotic stability

KW - Exchange of intersection and supremum

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KW - Nondegenerate Markov chains

KW - Nonlinear filtering

KW - Unique ergodicity

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Ergodicity and stability of the conditional distributions of nondegenerate markov chains

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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