Some Study Of Markov Chains And Markov Chain Fields

In the study of the strong law of large numbers in probability theory, non-homogeneous Markov chains investigated by many scholars have been correspondingly restricted in their works. But there is, little discussion about the Markov chains taking values in continuous states. In this paper, a class of strong limit theorem about the non-homogeneous Markov chain taking values in the continuous state are obtained in virtue of the notion of likelihood ratio and martingale convergence theorem.With the development of information theory, tree models have stimulated extensive interest in physics, probability theory and information theory in recent years. The study of Shannon-McMillan theorem in information theory is ever becoming a focus among many scholars. Recently, professor Liu Wen has introduced his analytic technique to the study of limit theorem of Markov chain fields on trees. In this paper, using the method of analytic technique of professor Liu Wen, we study the properties of the Markov chain fields on generalized Cayley trees and Bethe trees. Then some strong limit theorems and inductions on the frequencies of states and ordered couples of states are obtained, and a strong deviation theorem which is a new kind of theorem of studying arbitrary random fields on generalized Cayley trees and Bethe trees relative to Markov chain fields on it by introducing a measure of relative logarithmic likelihood ratio is also given.