### Abstract

We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365-393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space.

Original language | English (US) |
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Pages (from-to) | 1876-1925 |

Number of pages | 50 |

Journal | Annals of Probability |

Volume | 37 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2009 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Asymptotic stability
- Exchange of intersection and supremum
- Hidden Markov models
- Markov chain in random environment
- Nonlinear filtering
- Tail s-field
- Weak ergodicity