Strong Limit Properties For Generalized Non-Symmetric Markov Chain Fields On Trees

Random fields on trees are applications on trees of theory of random process-anew math model, which developed from coding and encoding problem in informationtheory. Assuming there is a sequence of｛X_n｝, whether the appearing frequency ofstate and state couple obey the strong law of large numbers is the key of a goodcoding and encoding method, so this domain is always being a researching emphasesfor many scholars. Thirty years ago, when random fields came into being. It 's asubject of intersection of Probability and Statistical Physics. Random fields, togetherwith other branches of probabilistic Physics, stand for an important aspect of a trend,which is the interpenetration of Math. and Phys.. In this paper, we have studied somestrong limit properties for generalized non-symmetric Markov chain field on trees.With the development of the information theory, the tree model has drawn increasinginterest from specialist in physics,probability and information theory. TheShannon-McMillan theorem in information theory is ever becoming a focus amongmany scholars. Professor Yang Weiguo and his associates extend some strong limittheorems and Shannon-McMillan theorem for classical Markov chains to Markovchains on Bethe trees and Cayley trees recently. But for non-homogeneous Markovchain fields, they only study the even-odd Markov chain. In this paper, we havestudied some strong limit properties for generalized non-symmetric Markov chainfield on trees. The definition of generalized non-symmetric Markov chain fields onCayley tree is given. By applying the Doob's Martingale convergence theorem andsome special inequalities, the strong limit theorem for generalized non-symmetricMarkov chain fields on Cayley tree are studied. A local convergence theorem and thestrong law of large numbers for the frequencies of occurrence of states and orderedcouples of states for generalized non-symmetric Markov chain fields on Cayley treeare obtained. Then, Shannon-McMillan Theorem with a.s. convergence forgeneralized non-symmetric Markov chain fields on Cayley tree is proved. Someresults about the Markov chain fields are extended to generalized non-symmetricMarkov chain fields on Cayley tree.