This paper deals with the problem of modeling data generated by an ergodic stochastic process as the output of a hidden Markov model (HMM). More specifically, we consider the problem of fitting a parametric family of HMM with continuous output to an ergodic stochastic process with continuous values, which does not necessarily belong to the family. In this context, we derive the main asymptotic results: almost sure consistency of the maximum likelihood estimator, asymptotic normality of the estimation error and the exact rates of almost sure convergence.