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

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

2008-01-28

arxiv.org/abs/0801.4366v2

Annals of Probability 2009, Vol. 37, No. 5, 1876-1925

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP448 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

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 $\sigma$-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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