# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the maximum entropy principle for uniformly ergodic Markov chains. (English) Zbl 0691.60023

Let $\left\{{X}_{j}:$ $j\in {N}_{0}\right\}$, ${N}_{0}=N\cup \left\{0\right\}$, be a Markov chain on (${\Omega }$,$𝒜,P\right)$ with Polish state space E, and let ${L}_{n}={n}^{-1}{\sum }_{j=1}^{n}{\delta }_{{X}_{j}}$. Let H be some function defined on the set of probability measures on E with values in [- $\infty ,\infty \right)$ which is nice enough. Transformed laws are defined by

${\stackrel{^}{P}}_{n}\left(A\right)=\left({\int }_{A}exp\left\{nH\left({L}_{n}\right)\right\}dP\right){\left({\int }_{{\Omega }}exp\left\{nH\left({L}_{n}\right)\right\}dP\right)}^{-1},\phantom{\rule{1.em}{0ex}}A\in 𝒜·$

The possible limit laws of $\left\{{X}_{j}:$ $j\in {N}_{0}\right\}$ under ${\stackrel{^}{P}}_{n}$ are described. The main assumption is that $\left\{{X}_{j}:$ $j\in {N}_{0}\right\}$ 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:
 60F10 Large deviations 60J05 Discrete-time Markov processes on general state spaces 60F05 Central limit and other weak theorems 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)