## Exponential forgetting and geometric ergodicity in hidden Markov models.(English)Zbl 0941.93053

This paper considers 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, it is proved 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.

### MSC:

 93E11 Filtering in stochastic control theory 15B52 Random matrices (algebraic aspects)
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