# On the maximum entropy principle for uniformly ergodic Markov chains

Bolthausen, E; Schmock, U (1989). On the maximum entropy principle for uniformly ergodic Markov chains. Stochastic Processes and their Applications, 33(1):1-27.

## 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.

## 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.

## Statistics

### Citations

15 citations in Web of Science®
11 citations in Scopus®

### Altmetrics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics maximum entropy; large deviations; Markov chains; variational problem; weak convergence English 1989 04 Nov 2009 14:31 05 Apr 2016 13:29 Elsevier 0304-4149 https://doi.org/10.1016/0304-4149(89)90063-X http://www.ams.org/mathscinet-getitem?mr=1027105http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0691.60023