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The Generalized Entropy Ergodic Theorem with Two Types of Convergence for M-th-Order Nonhomogeneous Hidden Markov Models

Author

Listed:
  • Qifeng Yao

    (Nanjing University of Science and Technology)

  • Longsheng Cheng

    (Nanjing University of Science and Technology)

  • Wenhe Chen

    (Nanjing University of Science and Technology)

  • Ting Mao

    (Nanjing University of Science and Technology)

  • Zhipeng Chang

    (Anhui University of Technology)

Abstract

A nonhomogeneous hidden Markov model (NHMM) consists of an unobservable nonhomogeneous Markov chain and an observable stochastic process. If the Markov assumption is changed to the condition that each hidden state in each step is related to the previous m states, then the extended model is called an m-th-order NHMM. Considering the calculation of entropy rate in the case of delayed averages, the generalized entropy ergodic theorem with almost-everywhere and $$\mathcal {L}_1$$ L 1 convergence for m-th-order NHMMs is studied in this paper. The establishment of some inequalities supports the proof of convergence for two types, and the obtained results maintain strict consistency with the corresponding classical results.

Suggested Citation

  • Qifeng Yao & Longsheng Cheng & Wenhe Chen & Ting Mao & Zhipeng Chang, 2025. "The Generalized Entropy Ergodic Theorem with Two Types of Convergence for M-th-Order Nonhomogeneous Hidden Markov Models," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-13, March.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:1:d:10.1007_s10959-024-01386-6
    DOI: 10.1007/s10959-024-01386-6
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    References listed on IDEAS

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    1. Zhongzhi Wang & Weiguo Yang, 2016. "The Generalized Entropy Ergodic Theorem for Nonhomogeneous Markov Chains," Journal of Theoretical Probability, Springer, vol. 29(3), pages 761-775, September.
    2. Zhiyan Shi & Weiguo Yang & Bei Wang, 2016. "A class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(23), pages 7027-7039, December.
    3. Zhiyan Shi & Xinyue Xi & Qingpei Zang, 2024. "The generalized entropy ergodic theorem for Markov chains indexed by a spherically symmetric tree," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 53(6), pages 2178-2193, March.
    4. Guoqing Yang & Weiguo Yang & Xiaotai Wu, 2017. "The strong laws of large numbers for countable non homogeneous hidden Markov models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(17), pages 8808-8819, September.
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