Recursive estimation in hidden Markov models

We consider a hidden Markov model (HMM) with multidimensional observations, and where the coefficients (transition probability matrix, and observation conditional densities) depend on some unknown parameter. We study the asymptotic behaviour of two recursive estimators, the recursive maximum likelihood estimator (RMLE), and the recursive conditional least squares estimator (RCLSE), as the number of observations increases to infinity. Firstly, we exhibit the contrast functions associated with the two non-recursive estimators, and we prove that the recursive estimators converge a.s. to the set of stationary points of the corresponding contrast function. Secondly, we prove that the two recursive estimators are asymptotically normal.