Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T10:32:52.810Z Has data issue: false hasContentIssue false
  • English
  • Français

A CLASS OF SMALL DEVIATION THEOREMS FOR FUNCTIONALS OF RANDOM FIELDS ON A TREE WITH UNIFORMLY BOUNDED DEGREE IN RANDOM ENVIRONMENT

Published online by Cambridge University Press:  24 August 2020

Zhiyan Shi Open the ORCID record for Zhiyan Shi [Opens in a new window]  and
Chengjun Ding
Zhiyan Shi
Affiliation:
School of Mathematical Science, Jiangsu University, Zhenjiang 212013, China E-mail: shizhiyan1984@126.com
Chengjun Ding
Affiliation:
School of Mathematical Science, Jiangsu University, Zhenjiang 212013, China E-mail: shizhiyan1984@126.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we mainly study a class of small deviation theorems for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment. Firstly, we give the definition of Markov chains indexed by a tree with uniformly bounded degree in random environment. Then, we introduce the some lemmas which are the basis of the results. Finally, a class of small deviation theorems for functionals of random fields on a tree with uniformly bounded degree in Markovian environment is established.

Keywords

random environmentsmall deviation theoremsstrong limit theoremstreeuniformly bounded degree

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 1 , January 2022 , pp. 169 - 183
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

References

Benjamini, I. & Peres, Y. (1994). Markov chains indexed by trees. The Annals of Probability 22: 219–243.10.1214/aop/1176988857CrossRefGoogle Scholar
Berger, T. & Ye, Z. (1990). Entropic aspects of random fields on trees. IEEE Transactions on Information Theory 36: 1006–1018.CrossRefGoogle Scholar
Cogburn, R. (1984). The ergodic theory of Markov chains in random environments. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 66: 109–128.CrossRefGoogle Scholar
Cogburn, R. (1990). On direct convergence and periodicity for transition probabilities of Markov chains in random environments. The Annals of Probability 18(2): 642–654.10.1214/aop/1176990850CrossRefGoogle Scholar
Cogburn, R. (1991). On the central limit theorem for Markov chains in random environments. The Annals of Probability 19(2): 587–604.CrossRefGoogle Scholar
Dang, H., Yang, W.G., & Shi, Z.Y. (2015). The strong law of large numbers and the entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree. IEEE Transactions on Information Theory 61(4): 1640–1648.CrossRefGoogle Scholar
Hu, D.H. & Hu, S.Y. (2009). On Markov chains in space-time random environments. Acta Mathematica Scientia 29B(1): 1–10.Google Scholar
Huang, H.L. (2017). The asymptotic behavior for Markov chains in a finite i.i.d random environment indexed by Cayley trees. Filomat 31(2): 273–283.10.2298/FIL1702273HCrossRefGoogle Scholar
Huang, H.L. (2019). The generalized entropy theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Probability in the Engineering and Informational Sciences. doi:10.1017/S0269964818000554.Google Scholar
Huang, H.L. & Yang, W.G. (2008). Strong law of large number for Markov chains indexed by an infinite tree with uniformly bounded degree. Science in China Series A: Mathematics 51(2): 195–202.CrossRefGoogle Scholar
Liu, W. & Wang, L.Y. (2003). The Markov approximation of the random fields on Cayley trees and a class of small deviation theorems. Acta Mathematica Scientia 63(2): 113–121.Google Scholar
Liu, W. & Yang, W.G. (2000). The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems. Stochastic Processes and their Applications 89(1): 117–130.CrossRefGoogle Scholar
Liu, W., Ma, C.Q., Li, Y.Q., & Wang, S.M. (2015). A strong limit theorem for the average of ternary functions of Markov chains in bi-infinite random environments. Statistics & Probability Letters 63: 113–121.CrossRefGoogle Scholar
Nawrotzki, K. (1981). Discrete open system or Markov chains in a random environment I. Journal of Information Process, Cybernet 17: 569–599.Google Scholar
Nawrotzki, K. (1982). Discrete open system or Markov chains in a random environment II. Journal of Information Process, Cybernet 18: 83–98.Google Scholar
Pemantle, R. (1992). Antomorphism invariant measure on trees. The Annals of Probability 20: 1549–1566.10.1214/aop/1176989706CrossRefGoogle Scholar
Peng, W.C. (2015). Strong law of large numbers for Markov chains indexed by spherically symmetric tree. Probability in the Engineering and Informational Sciences 29(3): 473–481.Google Scholar
Peng, W.C., Yang, W.G., & Wang, B. (2010). A class of small deviation theorems for functionals of random fields on a homogeneous tree. Journal of Mathematical Analysis and Applications 361(2): 293–301.CrossRefGoogle Scholar
Shi, Z.Y. & Yang, W.G. (2017). The definition of tree-indexed Markov chains in random environment and their existence. Communications in Statistics – Theory and Methods 46(16): 7934–7941.CrossRefGoogle Scholar
Shi, Z.Y., Yang, W.G., & Wang, B. (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 45(23): 7027–7039.CrossRefGoogle Scholar
Shi, Z.Y., Zhong, P.P., & Fan, Y. (2018). The Shannon-McMillan theorem for Markov chains indexed by a Cayley tree in random environment. Probability in the Engineering and Informational Sciences 32: 626–639.10.1017/S0269964817000444CrossRefGoogle Scholar
Shi, Z., Zhou, H., & Fan, Y. (2019). A class of small deviation theorems for functionals of random fields on double Cayley tree in random environment. Stochastics. doi:10.1080/17442508. 2019.1691209.Google Scholar
Shi, Z., Yang, W., & Tang, Y. (2020). Equivalent properties for the bifurcating Markov chains indexed by a binary tree. Communications in Statistics – Theory and Methods. doi:10.1080/03610926.2020.1742923.Google Scholar
Shi, Z., Liu, C., Fan, Y., Bao, D., & Chen, Y. (2020). Some generalized strong limit theorems for Markov chains in bi-infinite random environments. Communications in Statistics – Theory and Methods. doi:10.1080/03610926.2020.1744655.Google Scholar
Wang, B. & Shi, Z.Y. (2013). The strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains. Journal of Inequalities and Applications 2013(1): 1–8.10.1186/1029-242X-2013-462CrossRefGoogle Scholar
Wang, S. & Yang, W.G. (2013). A class of small deviation theorems for random fields on a uniformly bounded tree. Journal of Inequalities and Applications 2013(1): 1–12.Google Scholar
Wang, B., Yang, W.G., & Shi, Z.Y. (2012). Strong laws of large numbers for countable Markov chains indexed by a Cayley tree. Scientia Sinica Mathematica 42(10): 1031–1036.Google Scholar
Yang, W.G. & Liu, W. (2000). Strong law of large numbers for Markov chains fields on a Bethe tree. Statistics & Probability Letters 49: 245–250.10.1016/S0167-7152(00)00053-5CrossRefGoogle Scholar
Yang, W.G. & Ye, Z. (2007). The asymptotic equipartition property for Markov chains indexed by a homogeneous tree. IEEE Transactions on Information Theory 53(9): 3275–3280.CrossRefGoogle Scholar
Ye, Z. & Berger, T. (1996). Ergodic, regulary and asymptotic equipartition property of random fields on trees. Journal of Combinatorics, Information & System Sciences 21: 157–184.Google Scholar
Zhao, M., Shi, Z., Yang, W., & Wang, B. (2020). A class of strong deviation theorems for the sequence of real valued random variables with respect to continuous-state nonhomogeneous Markov chains. Communications in Statistics – Theory and Methods. doi:10.1080/03610926.2020.1734838.Google Scholar