Some Strong Limit Properties For Markov Fields On A Special Non-Homogeneous Trees

Random fields on trees are applications of Random Process theory on trees,Markov chain fields are the Markov chains indexed by tree index.Markov random fields theory is one of branches of probability theory developed recently, and the strong limit theorems of Markov chains fields is also its important compontent part.Professor Liu Wen firstly introduced his analytic techniques to the study the limit theorems of Markov chain fields on trees.In the chapter two of the paper,we firstly construct a class of special non-homogeneous trees T_{α0,α1,…,αr-1},using the methods of analytic techniques of Liu Wen,we study the limit properties of Markov chain fields on the special non-homogeneous trees T_{α0,α1,…,αr-1},we obtained some strong limit theorems and introductions on the frequencies of states,the ordered couple states,and the r-tuple of states set(i₀,i₁…,i_r-1).The recurrence of Markov chain fields on trees gives us a new description of the rules of appearence of state k in another attitudes.In the first section of the chapter three,using the Martingale methods,we study the recurrence of Markov chain fields on trees,and we obtain a sufficient condition for the non-recurrence,the strong recurrence of the Markov chain on trees on condition that the state set is finite.In the second section of the chapter three,we study the ray recurrence of Markov chain fields on Cayley trees without confining the initial state x₀.In the last chapter,we summed up what we have done in the paper.