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Some limit properties for Markov chains indexed by a homogeneous tree

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  • Yang, Weiguo

Abstract

We first study a local convergence theorem for countable Markov chains indexed by a homogeneous tree. As corollaries, we obtain some limit theorems for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a homogeneous tree. Finally, we obtain the strong law of large numbers and Shannon-McMillan theorem for finite Markov chains indexed by a homogeneous tree. In the proof, a new technique for establishing the strong limit theorems in probability theory is applied.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:65:y:2003:i:3:p:241-250
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    References listed on IDEAS

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    1. Yang, Weiguo & Liu, Wen, 2000. "Strong law of large numbers for Markov chains field on a Bethe tree," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 245-250, September.
    2. Liu, Wen & Yang, Weiguo, 2000. "The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 117-130, September.
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    Cited by:

    1. 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.
    2. Li, Gaorong & Chen, Shuang & Zhang, Junhua, 2009. "A class of random deviation theorems and the approach of Laplace transform," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 202-210, January.
    3. Delmas, Jean-François & Marsalle, Laurence, 2010. "Detection of cellular aging in a Galton-Watson process," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2495-2519, December.
    4. 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.

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