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Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree. (English) Zbl 1141.60011
Summary: In this paper, we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree. The results generalize the analogous results on a homogeneous tree.

60F15Strong limit theorems
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
Full Text: DOI
[1] Benjamini I, Peres Y. Markov chains indexed by trees. Ann Probab, 22: 219--243 (1994) · Zbl 0793.60080 · doi:10.1214/aop/1176988857
[2] Kemeny J G, Snell J L, Knapp A W. Denumberable Markov Chains. New York: Springer, 1976 · Zbl 0348.60090
[3] Spitzer F. Markov random fields on an infinite tree. Ann Probab, 3: 387--398 (1975) · Zbl 0313.60072 · doi:10.1214/aop/1176996347
[4] Berger T, Ye Z. Entropic aspects of random fields on trees. IEEE Trans Inform Theory, 36: 1006--1018 (1990) · Zbl 0738.60100 · doi:10.1109/18.57200
[5] Ye Z, Berger T. Ergodic, regulary and asymptotic equipartition property of random fields on trees. J Combin Inform System Sci, 21: 157--184 (1996) · Zbl 0953.94008
[6] Ye Z, Berger T. Information Measures for Discrete Random Fields. Beijing: Science Press, 1998 · Zbl 0997.94532
[7] Pemantle R. Antomorphism invariant measure on trees. Ann Prob, 20: 1549--1566 (1992) · Zbl 0760.05055 · doi:10.1214/aop/1176989706
[8] Yang W G, Liu W. Strong law of large numbers for Markov chains fields on a Bethe tree. Statist Prob Lett, 49: 245--250 (2000) · Zbl 0981.60064 · doi:10.1016/S0167-7152(00)00053-5
[9] Takacs C. Strong law of large numbers for branching Markov chains. Markov Proc Related Fields, 8: 107--116 (2001) · Zbl 1042.60014
[10] Liu W, Yang W G. A extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains. Stochastic Process Appl, 61: 129--145 (1996) · Zbl 0861.60042 · doi:10.1016/0304-4149(95)00068-2
[11] Liu W, Yang W G. The Markov approximation of the sequences of N-valued random variables and a class of deviation theorems. Stochastic Process Appl, 89: 117--130 (2000) · Zbl 1051.94005 · doi:10.1016/S0304-4149(00)00016-8
[12] Yang W G. Some limit properties for Markov chains indexed by a homogeneous tree. Statist Prob Lett, 65: 241--250 (2003) · Zbl 1068.60045 · doi:10.1016/j.spl.2003.04.001