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Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1431–1452 | Cite as

Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains

  • Valérie GirardinEmail author
  • Loick Lhote
  • Philippe Regnault
Article
  • 22 Downloads

Abstract

Closed-form expressions for generalized entropy rates of Markov chains are obtained through pertinent averaging. First, the rates are expressed in terms of Perron-Frobenius eigenvalues of perturbations of the transition matrices. This leads to a classification of generalized entropy functionals into five exclusive types. Then, a weighted expression is obtained in which the associated Perron-Frobenius eigenvectors play the same role as the stationary distribution in the well-known weighted expression of Shannon entropy rate. Finally, all terms are shown to bear a meaning in terms of dynamics of an auxiliary absorbing Markov chain through the notion of quasi-limit distribution. Illustration of important properties of the involved spectral elements is provided through application to binary Markov chains.

Keywords

Entropy rate Ergodic Markov chain Entropy distribution Quasi-limit distribution 

Mathematics Subject Classification (2010)

94A17 60J10 15B48 47B65 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques N. Oresme UMR6139Université de Caen NormandieCaenFrance
  2. 2.ENSICAEN, GREYC, UMR 6072Université de Caen NormandieCaenFrance
  3. 3.Laboratoire de Mathématiques de Reims, FRE 2011Université de Reims Champagne-ArdenneReims Cedex 2France

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