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On the maximum entropy principle for uniformly ergodic Markov chains. (English) Zbl 0691.60023

Let {X j : jN 0 }, N 0 =N{0}, be a Markov chain on (Ω,𝒜,P) with Polish state space E, and let L n =n -1 j=1 n δ X j . Let H be some function defined on the set of probability measures on E with values in [- ,) which is nice enough. Transformed laws are defined by

P ^ n (A)=( A exp{nH(L n )}dP)( Ω exp{nH(L n )}dP) -1 ,A𝒜·

The possible limit laws of {X j : jN 0 } under P ^ n are described. The main assumption is that {X j : jN 0 } is uniformly ergodic. Roughly speaking, the limit laws are mixtures of Markov chains minimizing a certain free energy. The method of proof strongly relies on large deviation techniques.

Reviewer: B.Kryžienė

MSC:
60F10Large deviations
60J05Discrete-time Markov processes on general state spaces
60F05Central limit and other weak theorems
60J10Markov chains (discrete-time Markov processes on discrete state spaces)