We study strong limit theorems for a sequence of bivariate functions for an inhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states, ordered couples of states, and the conditional expectation of a bivariate function on a Cayley tree.
Similar content being viewed by others
References
W. Liu and W. G. Yang, “Some strong limit theorems for Markov chain fields on trees,” Probab. Eng. Inform. Sci., 18, 411–422 (2004).
W. G. Yang and Z. X. Ye, “The asymptotic equipartition property for nonhomogeneous Markov chains indexed by a homogeneous tree,” IEEE Trans. Inform. Theory, 53, 3275–3280 (2007).
F. Spitzer, “Markov random fields on an infinite tree,” Ann. Probab., 3, 387–398 (1975).
I. Benjamini and Y. Peres, “Markov chains indexed by trees,” Ann. Probab., 22, No. 3, 219–243 (1994).
T. Berger and Z. Ye, “Entropic aspects of random fields on trees,” IEEE Trans. Inform. Theory, 36, No. 5, 1006–1018 (1990).
R. Pemantle, “Automorphism invariant measures on trees,” Ann. Probab., 20, 1549–1566 (1992).
Z. Ye and T. Berger, Information Measure for Discrete Random Fields on Trees, Science, New York (1998).
W. G. Yang, “Some limit properties for Markov chains indexed by homogeneous tree,” Stat. Probab. Lett., 65, 241–250 (2003).
W. G. Yang and W. Liu, “Strong law of large numbers and Shannon–McMillan theorem for Markov chain fields on trees,” IEEE Trans. Inform. Theory, 48, 313–318 (2002).
Z. Y. Shi and W. G. Yang, “A limit property of random transition probability for a nonhomogeneous Markov chain indexed by a tree,” Acta Math. Appl. Sinica, 31, 648–653 (2008).
A. N. Kolmogorov, “On the logical foundation of probability theory,” Lect. Notes Math., 1021 (1982).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 10, pp. 1336–1351, October, 2011.
Rights and permissions
About this article
Cite this article
Wang, K. A class of strong limit theorems for inhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system. Ukr Math J 63, 1517–1533 (2012). https://doi.org/10.1007/s11253-012-0597-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0597-1