Some Strong Laws For Markov Chain Fields Indexed By A Nonhomogeneous Tree Of Module M

The stochastic fields on trees is an application of stochastic process theory on tree model, and Markov chain fields on trees is an extension of Markov chains on tree indexed sets. Moreover, stochastic process indexed by a tree has become a hot topic in the field of the probability theory in recent years. The limit theory on Markov chain fields is being studied as an important content of stochastic process indexed by a tree.In this paper, convergence almost everywhere was studied by constructing a non-negative super-martingale and using Doob's martingale convergence theo-rem. Then some strong laws for Markov chain fields on a kind of special non-homogeneous tree of module m were obtained.This paper includes six chapters:The first chapter is introduction, introducing the researching purpose and mean-ings of this paper, and the work that existed.The second chapter is preparative knowledge. We give the concept of the tree and the definition of a non-homogeneous tree of module m.In the third chapter, we give some limit theorems of Markov chain on a non-homogeneous tree of module m and the strong laws of large numbers about oc-curred frequency of state and states of ordered couples.In the forth chapter, we give a strong deviation theorem of continuous state Markov chain fileds on a non-homogeneous tree of module m.In the fifth chapter, on the premise that the observed chain is mutually inde-pendent, we give and proof the existence theorem of mutual information rate of hidden Markov model on a non-homogeneous tree of module m.In the last chapter, we sum up what we have done in this paper.