Identifiability conditions for partially-observed Markov chains

We consider parametric models of partially-observed bivariate Markov chains. If the model is well-specified, we show under quite general conditions that the limiting normalizedlog-likelihood is maximized only by parameters for which the stationarydistribution is the same as the one of the true parameter. This is a keyfeature for obtaining the consistencyof the Maximum Likelihood Estimators (MLE), in cases where the parameter maynot be identifiable. The specific cases of Hidden Markov Models and Observation-driven timeseries are investigated. In contrast with previous approaches, this result isestablished by relying on the unicity of the invariant distribution of theMarkov chain associated to the complete data, regardless its rate ofconvergence to the equilibrium.