### 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 Y_{0},.., 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) |
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Pages (from-to) | 1495-1540 |

Number of pages | 46 |

Journal | Annals of Applied Probability |

Volume | 22 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 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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## Cite this

*Annals of Applied Probability*,

*22*(4), 1495-1540. https://doi.org/10.1214/11-AAP800