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STRONG LAW OF LARGE NUMBERS FOR MARKOV CHAINS INDEXED BY SPHERICALLY SYMMETRIC TREES

Published online by Cambridge University Press:  16 April 2015

Peng Weicai ,
Yang Weiguo  and
Shi Zhiyan
Peng Weicai
Affiliation:
Department of Mathematics, Chaohu University, Chaohu, 238000, People's Republic of China E-mail: weicaipeng@126.com
Yang Weiguo
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang, 212013, People's Republic of China
Shi Zhiyan
Affiliation:
Faculty of Science, Jiangsu University, Zhenjiang, 212013, People's Republic of China
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Abstract

In this paper, we main consider spherically symmetric tree T. First, under the condition lim supn→∞ |T(n)|/|Ln|<∞, we investigate the strong law of large numbers (SLLNs) for T-indexed Markov chains on the nth level of T. Then, combining the Stolz theorem, we obtain the SLLNs on T. Finally, we get Shannon–McMillan theorem for T-indexed Markov chains. The obtained theorems are generalizations of some known results on Cayley tree TC, N and Bethe tree TB, N.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 3 , July 2015 , pp. 473 - 481
Copyright
Copyright © Cambridge University Press 2015 

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References

1. Benjamini, I. & Peres, Y. (1994). Markov chains indexed by trees. Annals of Probability 22: 219–243.Google Scholar
2. Berger, T. & Ye, Z. (1990). Entropic aspects of random fields on trees. IEEE Transaction on Information Theory 36: 1006–1018.Google Scholar
3. Huang, H.L. & Yang, W.G. (2008). Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree. Science in China Series A: Mathematics 51(2): 195–202.Google Scholar
4. Pemantle, R. (1992). Automorphism invariant measure on trees. Annals of Probability 20: 1549–1566.Google Scholar
5. Liu, W. & Yang, W.G. (2004). Some strong limit theorems for Markov chain fields on trees. Probability in the Engineering and Informational Sciences 18(03): 411–422.Google Scholar
6. Takacs, C. (2001). Strong law of large numbers for branching Markov chains. Markov Process Related Fields 8: 107–116.Google Scholar
7. Wang, K.K. & Zong, D.C. (2011). Some Shannon–McMillan approximation theorems for Markov chain field on the generalized Bethe tree. Journal of Inequalities and Applications 2011. 2011:470910 (23 February 2011).Google Scholar
8. 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, 2013:81 (28 February 2013).Google Scholar
9. Yang, W.G. (2003). Some limit properties for Markov chains indexed by a homogeneous tree. Statistics & Probability Letters 65: 241–250.Google Scholar
10. Yang, W.G. & Liu, W. (2001). Strong law of large numbers and Shannon–McMillan theorem for Markov chains field on Cayle tree. Acta Mathematica Scientia 21B(4): 495–502.Google Scholar
11. Yang, W.G. & Ye, Z. (2007). The Asympotoic equipartition property for nonhomogeneous Markov chains indexed by a Homogeneous tree. IEEE Transactions on Information Theory 53(9): 3275–3280.Google Scholar
12. Ye, Z. & Berger, T. (1996). Ergodic, regulary and asymptotic equipartition property of random fields on trees. Journal of Combinatorics, Information and System Science 21: 157–184.Google Scholar