The Annals of Probability

The stability of conditional Markov processes and Markov chains in random environments

Ramon van Handel

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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.

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Ann. Probab., Volume 37, Number 5 (2009), 1876-1925.

First available in Project Euclid: 21 September 2009

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Zentralblatt MATH identifier

Primary: 93E11: Filtering [See also 60G35]
Secondary: 60J05: Discrete-time Markov processes on general state spaces 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 93E15: Stochastic stability

Nonlinear filtering asymptotic stability hidden Markov models weak ergodicity tail σ-field exchange of intersection and supremum Markov chain in random environment


van Handel, Ramon. The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 (2009), no. 5, 1876--1925. doi:10.1214/08-AOP448.

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