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The Generalized Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree

Author

Listed:
  • Zhiyan Shi

    (Jiangsu University)

  • Zhongzhi Wang

    (Anhui University of Technology)

  • Pingping Zhong

    (Jiangsu University)

  • Yan Fan

    (Jiangsu University)

Abstract

In this paper, we study the generalized entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree. Firstly, by constructing a class of random variables with a parameter and the mean value of one, we establish a strong limit theorem for delayed sums of the bivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we prove the strong law of large numbers for the frequencies of occurrence of states of delayed sums and the generalized entropy ergodic theorem. As corollaries, we generalize some known results.

Suggested Citation

  • Zhiyan Shi & Zhongzhi Wang & Pingping Zhong & Yan Fan, 2022. "The Generalized Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1367-1390, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01117-1
    DOI: 10.1007/s10959-021-01117-1
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    References listed on IDEAS

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    1. Yang, Weiguo, 2003. "Some limit properties for Markov chains indexed by a homogeneous tree," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 241-250, November.
    2. Shi, Zhiyan & Yang, Weiguo, 2010. "Some limit properties for the mth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree," Statistics & Probability Letters, Elsevier, vol. 80(15-16), pages 1223-1233, August.
    3. Dong, Yan & Yang, Weiguo & Bai, Jianfang, 2011. "The strong law of large numbers and the Shannon–McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1883-1890.
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