Some Strong Limit Theorems For Markov Chains Indexed By A Non-homogeneous Tree

The combination of tree graphs model and Markov chains have produced a new mathematical theory system—tree indexed Markov chains.In recent years,it has been applied to many fields,such as probability theory,information theory and computer science,etc.As the basis and important branch of probability theory and statistics,strong limit theorem reveals the regularity of random phenomena when its observation times tend to infinity.Therefore,popularizing strong limit theorem to Markov chains indexed by a tree has great significance.Based on the classical limit theory,this paper extends it to more extensive cases,and gives detailed proof,which provides the theoretical basis for more practical problems modeling.As the basis of the research,the paper defines a non-homogeneous tree T.Under the definition,by constructing non-negative martingales and applying Doob's martingale convergence theorem,combining important inequality,properties of upper limit and transformation between sums and products of logarithmic function,etc,we give and prove three main theorems and a corollary for Markov chains indexed by a nonhomogeneous tree.Firstly,we introduce the concept of generalized gambling systems and give a strong limit theorem of generalized gambling systems for Markov chains indexed by a non-homogeneous tree.Then,we give and prove a strong limit property for double Markov chains indexed by a non-homogeneous tree.Finally,we introduce the concept of sample divergence and give a strong deviation theorem of sample divergence for double Markov chains indexed by a tree.Meanwhile,as a corollary,a property for double Markov chains indexed by a tree is given when the sample divergence is 0.