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SOME LIMIT THEOREMS OF DELAYED AVERAGES FOR COUNTABLE NONHOMOGENEOUS MARKOV CHAINS

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  24 September 2019

Pingping Zhong ,
Weiguo Yang ,
Zhiyan Shi Open the ORCID record for Zhiyan Shi [Opens in a new window]  and
Yan Zhang
Pingping Zhong
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang212013, China E-mail: shizhiyan1984@126.comJingjiang College of Jiangsu University, Zhenjiang 212013, China
Weiguo Yang
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang212013, China E-mail: shizhiyan1984@126.comJingjiang College of Jiangsu University, Zhenjiang 212013, China
Zhiyan Shi
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang212013, China E-mail: shizhiyan1984@126.comJingjiang College of Jiangsu University, Zhenjiang 212013, China
Yan Zhang
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang212013, China E-mail: shizhiyan1984@126.comJingjiang College of Jiangsu University, Zhenjiang 212013, China
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Abstract

The purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

Keywords

countable nonhomogeneous Markov chainsgeneralized C-strong ergodicitystrong law of large numbers of delayed averages

MSC classification

Primary: 60G60: Random fields
Secondary: 60F15: Strong theorems
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 297 - 315
Copyright
Copyright © Cambridge University Press 2019

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References

1.Bowerman, B., David, H.T., & Isaacson, D. (1977). The convergence of Cesàro averages for certain nonstationary Markov chains. Stochastic Processes and their Applications 5: 221–230.CrossRefGoogle Scholar
2.Chiang, T.S. & Chow, Y.S. (1989). A limit theorem for a class of inhomogeneous Markov processes. Annals of Probability 17: 1483–1502.CrossRefGoogle Scholar
3.Cover, T.M. & Thomas, J.A. (1991). Elements of Information Theory. New York: Wiley.CrossRefGoogle Scholar
4.Dietz, Z. & Sethuraman, S. (2005). Large deviations for a class of nonhomogeneous Markov chains. Annals of Applied Probability 15: 421–486.CrossRefGoogle Scholar
5.Isaacson, D. & Madsen, R. (1976). Markov Chains Theory and Application. New York: Wiley.Google Scholar
6.Isaacson, D. & Senta, E. (1982). Ergodicity for countable inhomogeneous Markov chains. Linear Algebra and its Applications 48: 37–44.CrossRefGoogle Scholar
7.Liu, G.X. & Liu, W. (1994). On the strong law of large numbers for functionals of countable nonhomogeneous Markov chains. Stochastic Processes and their Applications 50: 375–391.CrossRefGoogle Scholar
8.Liu, W. & Yang, W.G. (1996). An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains. Stochastic Processes and their Applications 61: 129–145.Google Scholar
9.Ma, H.L. & Yang, W.G. (2011). Erratum to “The asymptotic equipartition property for asymptotic circular Markov chains”. Probability in the Engineering and Informational Sciences 25(2): 265–267.CrossRefGoogle Scholar
10.Mukhamedov, F. (2013). The Dobrushin ergodicity coefficient and the ergodicity of noncommutative Markov chains. Journal of Mathematical Analysis and Applications 408: 364–373.CrossRefGoogle Scholar
11.Rosenblatt-Roth, M. (1964). Some theorems concerning the strong law of large numbers for non-homogeneous Markov chains. The Annals of Mathematical Statistics 35: 566–576.CrossRefGoogle Scholar
12.Tan, C.P. (1996). On the weak ergodicity of nonhomogeneous Markov chains. Statistics & Probability Letters 26: 293–295.CrossRefGoogle Scholar
13.Wang, Z.Z. & Yang, W.G. (2016). The generalized entropy ergodic theorem for nonhomogeneous Markov Chains. Journal of Theoretical Probability 29: 761–775.CrossRefGoogle Scholar
14.Waqas, A. & Yang, W.G. (2019). Generalized absolute mean strong ergodicity of nonhomogeneous Markov Chains. Pakistan Journal of Statistics 35: 53–63.Google Scholar
15.Yang, W.G. (1998). The asymptotic equipartition property for a nonhomogeneous Markov information source. Probability in the Engineering and Informational Sciences 12: 509–518.CrossRefGoogle Scholar
16.Yang, W.G. (2002). Convergence in the Cesáro sense and strong law of large numbers for countable nonhomogeneous Markov chains. Linear Algebra and its Applications 354: 275–286.CrossRefGoogle Scholar
17.Yang, W.G. (2009). Strong law of large numbers for countable nonhomogeneous Markov chains. Linear Algebra and its Applications 430: 3008–3018.CrossRefGoogle Scholar
18.Yang, W.G., Wang, B., & Shi, Z.Y. (2014). Strong law of large numbers for countable asymptotic circular Markov Chains. Communications in Statistics – Theory and Methods 43(18): 3943–3954.CrossRefGoogle Scholar
19.Zhong, P.P., Yang, W.G., & Liang, P.P. (2010). The asymptotic equipartition property for asymptotic circular Markov chains. Probability in the Engineering and Informational Sciences 24: 279–288.CrossRefGoogle Scholar