Asymptotics of maximum likelihood estimators based on Markov chain Monte Carlo methods

In many complex statistical models maximum likelihood estimators cannot be calculated. In the paper we solve this problem using Markov chain Monte Carlo approximation of the true likelihood. In the main result we prove asymptotic normality of the estimator, when both sample sizes (the initial and Monte Carlo one) tend to infinity. Our result can be applied to models with intractable norming constants and missing data models.