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.
Publié le : 2009-09-15
Classification:
Nonlinear filtering,
asymptotic stability,
hidden Markov models,
weak ergodicity,
tail σ-field,
exchange of intersection and supremum,
Markov chain in random environment,
93E11,
60J05,
62M20,
93E15
@article{1253539859,
author = {van Handel, Ramon},
title = {The stability of conditional Markov processes and Markov chains in random environments},
journal = {Ann. Probab.},
volume = {37},
number = {1},
year = {2009},
pages = { 1876-1925},
language = {en},
url = {http://dml.mathdoc.fr/item/1253539859}
}
van Handel, Ramon. The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab., Tome 37 (2009) no. 1, pp. 1876-1925. http://gdmltest.u-ga.fr/item/1253539859/