Strong Limit Theorems For Arbitrary Stochastic Process Indexed By A Tree And Arbitrary Stochastic Arrays

Random fields on trees are applications of theory of stochastic process on trees -a new math model,which developed from coding and encoding problem in information theory.Assuming there is a process{X_t,t∈T},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 a researching emphases for many scholars.Thirty years ago,when random fields came into being,it's a subject of intersection of Probability and Statistic Physics,together with other branches of probabilistic Physics,standing for the general trend of an important aspect of today's mathematics and physical mutual penetration.With the development of the information theory,the tree model has drawn increasing interest from specialist in physics,probability and information theory. Recently,Professor Yang Weiguo and his associates extend some strong limit theorems and Shannon-McMillan theorem for classical Markov chains to Markov chains indexed by Bethe trees and Cayley trees.In this paper,we first give the definition of arbitrary stochastic process indexed by a tree,and use the new method of studying strong limit theorems in Yang's paper-constructing a non-negative martingale to extend these results for Markov chains indexed by a tree to strong limit theorems for arbitrary stochastic process indexed by a tree.As corollaries,we obtain some strong limit theorems for special stochastic process indexed by a tree.Finally,we also study the strong limit theorem for arbitrary stochastic arrays.As corollaries,we obtain some results in Yang's papers.