On the maximum entropy principle for uniformly ergodic Markov chains

Abstract

For strongly ergodic discrete time Markov chains we discuss the possible limits as n→∞ of probability measures on the path space of the form exp(nH(Ln)) dP/Zn· Ln is the empirical measure (or sojourn measure) of the process, H is a real-valued function (possibly attaining −∞) on the space of probability measures on the state space of the chain, and Zn is the appropriate norming constant. The class of these transformations also includes conditional laws given Ln belongs to some set. The possible limit laws are mixtures of Markov chains minimizing a certain free energy. The method of proof strongly relies on large deviation techniques.