On the Maximum Entropy Principle for Uniformly Ergodic Markov Chains

Erwin Bolthausen and Uwe Schmock

Abstract: For strongly ergodic discrete-time Markov chains we discuss the possible limits as n tends to infinity of probability measures on the path space of the form exp(nH(L_n))dP/Z_n. L_n is the empirical measure (or sojourn measure) of the process, H is a real-valued function (possibly attaining minus infinity) on the space of probability measures on the state space of the chain, and Z_n is the appropriate norming constant. The class of these transformations also includes conditional laws given L_n 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.

Keywords: maximum entropy principle, large deviations, Markov chains, variational problem, weak convergence

2010 Mathematics Subject Classification:

• 60F05 Central limit and other weak theorems
• 60F10 Large deviations
• 60J05 Markov processes with discrete parameter

Reference: Stochastic Processes and their Applications 33 (1989) 1-27.

DOI: 10.1016/0304-4149(89)90063-X

Please send comments and suggestions to Uwe Schmock, email: schmock@fam.tuwien.ac.at. Last update: June 14, 2003