Maximum likelihood estimation for multivariate observations of Markov sources

Parameter estimation for multivariate functions of Markov chains, a class of versatile statistical models for vector random processes, is discussed. The model regards an ordered sequence of vectors as noisy multivariate observations of a Markov chain. Mixture distributions are a special case. The foundations of the theory presented here were established by Baum, Petrie, Soules, and Weiss. A powerful representation theorem by Fan is employed to generalize the analysis of Baum, {\em et al.} to a larger class of distributions.