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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, Zhenjiang212013, China E-mail: shizhiyan1984@126.com
Chengjun Ding
Affiliation:
School of Mathematical Science, Jiangsu University, Zhenjiang212013, China E-mail: shizhiyan1984@126.com
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Abstract

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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
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