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Mathematics of Control, Signals and Systems

, Volume 13, Issue 1, pp 63–93 | Cite as

Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models

  • Francçois Le Gland
  • Laurent Mevel

Abstract.

We consider a hidden Markov model with multidimensional observations, and with misspecification, i.e., the assumed coefficients (transition probability matrix and observation conditional densities) are possibly different from the true coefficients. Under mild assumptions on the coefficients of both the true and the assumed models, we prove that: (i) the prediction filter, and its gradient with respect to some parameter in the model, forget almost surely their initial condition exponentially fast, and (ii) the extended Markov chain, whose components are the unobserved Markov chain, the observation sequence, the prediction filter, and its gradient, is geometrically ergodic and has a unique invariant probability distribution.

Key words. HMM, Misspecified model, Prediction filter, Exponential forgetting, Geometric ergodicity, Product of random matrices. 

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Copyright information

© Springer-Verlag London Limited 2000

Authors and Affiliations

  • Francçois Le Gland
    • 1
  • Laurent Mevel
    • 1
  1. 1.IRISA/INRIA, Campus de Beaulieu, 35042 Rennes Cédex, France. {legland,lmevel}@irisa.fr.FR

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