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September 2009 The stability of conditional Markov processes and Markov chains in random environments
Ramon van Handel
Ann. Probab. 37(5): 1876-1925 (September 2009). DOI: 10.1214/08-AOP448


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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Ramon van Handel. "The stability of conditional Markov processes and Markov chains in random environments." Ann. Probab. 37 (5) 1876 - 1925, September 2009.


Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1178.93142
MathSciNet: MR2561436
Digital Object Identifier: 10.1214/08-AOP448

Primary: 93E11
Secondary: 60J05 , 62M20 , 93E15

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

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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