We consider a hidden Markov model with multidimensional observations, and with
misspecification, ie, 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 …

We consider an hidden Markov model with multidimensional observations, and with
misspeci cation, ie the assumed coe cients (transition probability matrix, and observation
conditional densities) are possibly di erent from the true coe cients. Under mild assumptions
on the coe cients of both the true and the assumed models, we prove that:(i) the prediction
lter, and its gradient wrt some parameter in the model, forget almost surely their initial
condition exponentially fast, and (ii) the extended Markov chain, whose components are: the …