On the maximum entropy principle for uniformly ergodic Markov chains (Q582683)

Let \(\{X_ j:\) \(j\in N_ 0\}\), \(N_ 0=N\cup \{0\}\), be a Markov chain on (\(\Omega\),\({\mathcal A},P)\) with Polish state space E, and let \(L_ n=n^{-1}\sum^{n}_{j=1}\delta_{X_ j}\). Let H be some function defined on the set of probability measures on E with values in [- \(\infty,\infty)\) which is nice enough. Transformed laws are defined by \[ \hat P_ n(A)=(\int_{A}\exp \{nH(L_ n)\}dP)(\int_{\Omega}\exp \{nH(L_ n)\}dP)^{-1},\quad A\in {\mathcal A}. \] The possible limit laws of \(\{X_ j:\) \(j\in N_ 0\}\) under \(\hat P_ n\) are described. The main assumption is that \(\{X_ j:\) \(j\in N_ 0\}\) 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.