Leroux's method for general hidden Markov models

The method introduced by Leroux [Maximum likelihood estimation for hidden Markov models, Stochastic Process Appl. 40 (1992) 127-143] to study the exact likelihood of hidden Markov models is extended to the case where the state variable evolves in an open interval of the real line. Under rather minimal assumptions, we obtain the convergence of the normalized log-likelihood function to a limit that we identify at the true value of the parameter. The method is illustrated in full details on the Kalman filter model.