Likelihood-look-ahead inference on the equilibrium distribution of Markov chains

We propose a method for statistical inference on the stationary probability measure of a Markov chain with general state space whose transition function belongs to a parametric family. We extend the look-ahead method introduced by Glynn and Henderson to this situation, using maximum likelihood estimation based on data from the observed process. We show the consistency and asymptotic normality of our estimator and construct confidence intervals for the values of the stationary distribution. We illustrate our results with simulation studies of the Lindley process and the AR(1) process.