Skip to main content
SpringerLink
Log in

Tree-Indexed Markov Chains in Random Environment and Some of Their Strong Limit Properties

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In this paper, the authors first introduce the tree-indexed Markov chains in random environment, which takes values on a general state space. Then, they prove the existence of this stochastic process, and develop a class of its equivalent forms. Based on this property, some strong limit theorems including conditional entropy density are studied for the tree-indexed Markov chains in random environment.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Algoet, P. H. and Cover, T. M., A sandwich proof of the Shannon-McMillan-Breiman theorem, Ann. Probab., 16(2), 1988, 899–909.

    Article  MathSciNet  Google Scholar 

  2. Barron, A. R., The strong ergodic theorem for densities: Generalized Shannon-McMillan-Breiman theorem, Ann. Probab., 13(4), 1985, 1292–1303.

    Article  MathSciNet  Google Scholar 

  3. Benjamini, I. and Peres, Y., Markov chains indexed by trees, Ann. Probab., 22(1), 1994, 219–243.

    Article  MathSciNet  Google Scholar 

  4. Berger, T. and Ye, Z., Entropic aspects of random fields on trees, IEEE T. Inform. Theory, 36(5), 1990, 1006–1018.

    Article  MathSciNet  Google Scholar 

  5. Breiman, L., The individual ergodic theorem of information theory, Ann. Math. Statist., 28(3), 1957, 809–811.

    Article  MathSciNet  Google Scholar 

  6. Chung, K. L., A note on the ergodic theorem of information theory. Ann. Math. Statist., 32(2), 1961, 612–614.

    Article  MathSciNet  Google Scholar 

  7. Chung, K. L., A Course in Probability Theory (3rd ed.), Elsevier, Singapore, 2001.

    Google Scholar 

  8. Cogburn, R., The ergodic theory of Markov chains in random environments, Z. Wahrsch. Verw. Gebiete., 66(1), 1984, 109–128.

    Article  MathSciNet  Google Scholar 

  9. Cogburn, R., On direct convergence and periodicity for transition probabilities of Markov chains in random environments, Ann. Probab., 18(2), 1990, 642–654.

    Article  MathSciNet  Google Scholar 

  10. Cogburn, R., On the central limit theorem for Markov chains in random environments, Ann. Probab., 19(2), 1991, 587–604.

    Article  MathSciNet  Google Scholar 

  11. Dang, H., Yang, W. and Shi, Z., The strong law of large numbers and the entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree, IEEE T. Inform. Theory, 61(4), 2015, 1640–1648.

    Article  MathSciNet  Google Scholar 

  12. Delmas, J. F. and Marsalle, L., Detection of cellular aging in a Galton-Watson process, Stoch. Proc. Appl., 120(12), 2010, 2495–2519.

    Article  MathSciNet  Google Scholar 

  13. Dembo, A., Mörters, A. P. and Sheffield, S., Large deviations of Markov chains indexed by random trees, Ann. I. H. Poincare-Probab. Statist., 41(6), 2005, 971–996.

    Article  MathSciNet  Google Scholar 

  14. Dong, Y., Yang, W. and Bai, J., The strong law of large numbers and the Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree, Stat. Probabil. Lett., 81(12), 2011, 1883–1890.

    Article  MathSciNet  Google Scholar 

  15. Guyon, J., Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, Ann Appl Probab, 17(5–6), 2007, 1538–1569.

    MathSciNet  MATH  Google Scholar 

  16. Hu, D. and Hu, X., On Markov chains in space-time random environments, Acta. Math. Sci., 29B(1), 2009, 1–10.

    MathSciNet  MATH  Google Scholar 

  17. Huang, H. and Yang, W., Strong law of large number for Markov chains indexed by an infinite tree with uniformly bounded degree, Sci. China Ser. A., 51(2), 2008, 195–202.

    Article  MathSciNet  Google Scholar 

  18. Liu, W., Ma, C., Li, Y. and Wang, S., A strong limit theorem for the average of ternary functions of Markov chains in bi-infinite random environments, Stat. Probabil. Lett., 100, 2015, 12–18.

    Article  MathSciNet  Google Scholar 

  19. Liu, W. and Yang, W., An extension of Shannon-Mcmillan theorem and some limit properties for nonhomogeneous Markov chains, Stoch. Proc. Appl., 61(1), 1996, 129–145.

    Article  MathSciNet  Google Scholar 

  20. McMillan, B., The basic theorems of information theory, Ann. Math. Statist, 24, 1953, 196–219.

    Article  MathSciNet  Google Scholar 

  21. Nawrotzki, K., Discrete open systems or Markov chains in a random environment, I, J. Inform. Process Cybernet, 17, 1981, 569–599.

    MathSciNet  MATH  Google Scholar 

  22. Nawrotzki, K., Discrete open systems or Markov chains in a random environment, II, J. Inform. Process Cybernet, 18, 1982, 83–98.

    MathSciNet  MATH  Google Scholar 

  23. Pemantle, R., Automorphism invariant measure on trees, Ann Probab, 20(3), 1992, 1549–1566.

    Article  MathSciNet  Google Scholar 

  24. Peng, W., Yang, W. and Shi, Z., Strong law of large numbers for Markov chains indexed by spherically symmetric trees, Probab. Eng. Inform. Sc., 29(3), 2015, 473–481.

    Article  MathSciNet  Google Scholar 

  25. Shannon, C., A mathematical theory of communication, Bell Syst. Tech. J., 27, 1948, 379–423, 623–656.

    Article  MathSciNet  Google Scholar 

  26. Shi, Z., Wang, Z., Zhong, P., et al., The Generalized Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree, Journal of Theoretical Probability, https://doi.org/10.1007/s10959-021-01117-1.

  27. Shi, Z. and Yang, W., The definition of tree-indexed Markov chains in random environment and their existence, Commun. Stat-Theor. M., 46(16), 2017, 7934–7941.

    Article  MathSciNet  Google Scholar 

  28. Shi, Z., Zhong, P. and Fan, Y., The Shannon-McMillan theorem for Markov chains indexed by a Cayley tree in random environment, Probab. Eng. Inform. Sc., 32, 2018, 626–639.

    Article  MathSciNet  Google Scholar 

  29. Yang, J. and Yang, W. G., The generalized entropy ergodic theorem for nonhomogeneous Markov chains indexed by a Cayley tree, Chin Ann Math, Ser. A, 41(1), 2020, 99–114 (In Chinese).

    MathSciNet  MATH  Google Scholar 

  30. Yang, W., Some limit properties for Markov chains indexed by a homogeneous tree, Stat. Probabil. Lett., 65(3), 2003, 241–250.

    Article  MathSciNet  Google Scholar 

  31. Yang, W. and Liu, W., Strong law of large numbers and Shannon-McMillan theorem for Markov chain fields on trees, IEEE T. Inform. Theory, 48(1), 2002, 313–318.

    Article  MathSciNet  Google Scholar 

  32. Yang, W. and Liu, W., The asymptotic equipartition property for Mth-order nonhomogeneous Markov information sources, IEEE T. Inform. Theory, 50(12), 2004, 3326–3330.

    Article  Google Scholar 

  33. Ye, Z. and Berger, T., Ergodic, regulary and asymptotic equipartition property of random fields on trees, J. Combin. Inform. System Sci., 21(2), 1996, 157–184.

    MathSciNet  MATH  Google Scholar 

  34. Ye, Z. and Berger, T., Information Measures for Discrete Random Fields, Science, Beijing, 1998.

    MATH  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the editors and reviewers for their helpful and important comments, especially during the time with COVID19 pandemic.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

    Zhiyan Shi & Weiguo Yang

  2. Faculty of Sciences, Wuxi University, Wuxi, Jiangsu, 214105, China

    Bei Wang

  3. School of Mathematics & Physics Science and Engineering, Anhui University of Technology, Ma’anshan, Anhui, 243002, China

    Zhongzhi Wang

Authors
  1. Zhiyan Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weiguo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhongzhi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Zhiyan Shi, Bei Wang or Zhongzhi Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11571142, 11971197, 11601191).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, Z., Wang, B., Yang, W. et al. Tree-Indexed Markov Chains in Random Environment and Some of Their Strong Limit Properties. Chin. Ann. Math. Ser. B 43, 621–642 (2022). https://doi.org/10.1007/s11401-022-0349-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-022-0349-y

Keywords

2000 MR Subject Classification

Search

Navigation