A class of small deviation theorems for Markov chains in bi-infinite random environment

References

• Cogburn, R. 1980. Markov chains in random environments: The case of Markovian environments. The Annals of Probability 8 (5):908–16. doi:10.1214/aop/1176994620.
   Web of Science ®Google Scholar
• Cogburn, R. 1984. The ergodic theory of Markov chains in random environments. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 66 (1):109–28. doi:10.1007/BF00532799.
   Google Scholar
• Cogburn, R. 1990. On direct convergence and periodicity for transition probabilities of Markov chains in random environments. The Annals of Probability 18 (2):642–54. doi:10.1214/aop/1176990850.
   Web of Science ®Google Scholar
• Cogburn, R. 1991. On the central limit theorem for Markov chains in random environments. The Annals of Probability 19 (2):587–604. doi:10.1214/aop/1176990442.
   Web of Science ®Google Scholar
• Hu, D. H., and S. Y. Hu. 2009. On Markov chains in space-time random environments. Acta Mathematica Sinica 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):273–83. doi:10.2298/FIL1702273H.
   Web of Science ®Google Scholar
• Li, Y. Q. 2001. Recurrence and invariant measure of Markov chains in bi-infinite random environments. Science in China Series A: Mathematics 44 (10):1294–99. doi:10.1007/BF02877018.
   Google Scholar
• Liu, W., C. Q. Ma, Y. Q. Li, and S. M. Wang. 2015. A strong limit theorem for the average of ternary functions of Markov chains in bi-infinite random environments. Statistics & Probability Letters 100:12–18. doi:10.1016/j.spl.2015.01.029.
   Web of Science ®Google Scholar
• Liu, W., and L. Y. Wang. 2003. The Markov approximation of the random fields on Cayley trees and a class of small deviation theorems. Statistics & Probability Letters 63 (2):113–21. doi:10.1016/S0167-7152(03)00058-0.
   Web of Science ®Google Scholar
• Liu, W., and W. G. Yang. 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):117–30. doi:10.1016/S0304-4149(00)00016-8.
   Web of Science ®Google Scholar
• Nawrotzki, K. 1981. Discrete open system or Markov chains in a random environment I. Journal of Information Processing and Cybernetics 17:569–99.
   Google Scholar
• Nawrotzki, K. 1982. Discrete open system or Markov chains in a random environment II. Journal of Information Processing and Cybernetics 18:83–98.
   Google Scholar
• Peng, W. C., W. G. Yang, and B. Wang. 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. doi:10.1016/j.jmaa.2009.06.079.
   Web of Science ®Google Scholar
• Shi, Z. Y., D. Bao, Y. Fan, and B. H. Wu. 2019. The asymptotic equipartition property of Markov chains in single infinite Markovian environment on countable state space. Stochastics 91 (6):945–57. doi:10.1080/17442508.2019.1567730.
   Google Scholar
• Shi, Z. Y., J. L. Ji, and W. G. Yang. 2017. A class of small deviation theorem for the sequences of countable state random variables with respect to homogeneous Markov chains. Communications in Statistics - Theory and Methods 46 (14):6823–30. doi:10.1080/03610926.2015.1137594.
   Web of Science ®Google Scholar
• Shi, Z. Y., C. Liu, Y. Fan, and Y. Chen. 2020. Some generalized strong limit theorems for Markov chains in bi-infinite random environments. Communications in Statistics - Theory and Methods. Advance online publication. doi:10.1080/03610926.2020.1744655.
   Web of Science ®Google Scholar
• Shi, Z. Y., and W. G. Yang. 2017. The definition of tree-indexed Markov chains in random environment and their existence. Communications in Statistics - Theory and Methods 46 (16):7934–41. doi:10.1080/03610926.2016.1167907.
   Web of Science ®Google Scholar
• Shi, Z. Y., W. G. Yang, and B. Wang. 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):7027–39. doi:10.1080/03610926.2014.972746.
   Web of Science ®Google Scholar
• Wang, S., and W. G. Yang. 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