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On Hidden Markov Processes with Infinite Excess Entropy

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  • Łukasz Dębowski

    (Polish Academy of Sciences)

Abstract

We investigate stationary hidden Markov processes for which mutual information between the past and the future is infinite. It is assumed that the number of observable states is finite and the number of hidden states is countably infinite. Under this assumption, we show that the block mutual information of a hidden Markov process is upper bounded by a power law determined by the tail index of the hidden state distribution. Moreover, we exhibit three examples of processes. The first example, considered previously, is nonergodic and the mutual information between the blocks is bounded by the logarithm of the block length. The second example is also nonergodic but the mutual information between the blocks obeys a power law. The third example obeys the power law and is ergodic.

Suggested Citation

  • Łukasz Dębowski, 2014. "On Hidden Markov Processes with Infinite Excess Entropy," Journal of Theoretical Probability, Springer, vol. 27(2), pages 539-551, June.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:2:d:10.1007_s10959-012-0468-6
    DOI: 10.1007/s10959-012-0468-6
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    References listed on IDEAS

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    1. Bialek, William & Nemenman, Ilya & Tishby, Naftali, 2001. "Complexity through nonextensivity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 302(1), pages 89-99.
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