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

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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.

Original languageEnglish (US)
Pages (from-to)1876-1925
Number of pages50
JournalAnnals of Probability
Issue number5
StatePublished - Sep 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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


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