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

Author

Listed:
  • Valérie Girardin

    (Université de Caen Normandie)

  • Loick Lhote

    (Université de Caen Normandie)

  • Philippe Regnault

    (Université de Reims Champagne-Ardenne)

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.

Suggested Citation

  • Valérie Girardin & Loick Lhote & Philippe Regnault, 2019. "Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1431-1452, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9679-3
    DOI: 10.1007/s11009-018-9679-3
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    References listed on IDEAS

    as
    1. Valerie Girardin, 2004. "Entropy Maximization for Markov and Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 6(1), pages 109-127, March.
    2. Valérie Girardin & Philippe Regnault, 2016. "Escort distributions minimizing the Kullback–Leibler divergence for a large deviations principle and tests of entropy level," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 439-468, April.
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