Improved estimation of the exponential stability of the predictive filter in hidden Markov models

We consider finite state continuous read-out Hidden Markov Models. The exponential stability of the predictive filter was investigated by LeGalnd and Mevel when the transition probability matrix $Q$ of the underlying Markov chain is primitive. We carry out further investigation of this exponential stability. Two important applications are derived: the strong approximation result has been extended for HMMs with primitive transition probability matrices and the validity of the recursive estimation of HMMs with primitive transition probability matrices has been shown