Some Limit Properties For Markov Chains Indexed By A Tree

Random fields on trees are applications on trees of theory of random process-a new mathematical model,which developed from coding and encoding problem in information theory.Assuming there is a sequence of {X_n},whether the appearing frequency of state and state couple obey the strong law of large numbers is the key of a good coding and encoding method,so this domain is always being a researching emphases for many scholars Thirty years ago,when random fields came into being.It 's a subject of intersection of Probability and Statistical Physics.Random fields, together with other branches of probabilistic Physics,stand for an important aspect of a trend,which is the interpenetration of Math and Phys.Professor Yang Weiguo and his associates extend some strong limit theorems and Shannon-McMillan theorem for classical Markov chains to Markov chains on Bethe trees and Cayley trees recently.In this paper,we first give the definition of a general Cayley Tree and study a local convergence theorem for a finite second order Markov chain indexed by a general Cayley tree.As corollaries,we obtain some limit theorems for this Markov chain. Finally,we also obtain the strong law of large numbers(LLN) and Shannon-McMillan theorem for a class of finite second order Markov chain indexed by a general Cayley tree.At the same time,we study a limit property of the harmonic mean of random transition probability for a nonhomogenous Markov chains indexed by a tree.This result is an extension of the limit property of the harmonic mean of random transition probability for a nonhomogenous Markov chains.