A Class Of Strong Laws Of Large Numbers For Markov Chain Fields On A Tree

In recent years,the tree model has attracted a great deal of interest in relevant research fields with its foundation and particularity.Moreover,stochastic process indexed by a tree has gradually developed and is widely studied.The international probability theory community is paying more and more attention to the in-depth exploration of the topic of the strong law of large numbers.This paper mainly studies some strong law of large numbers of Markov chain fields on a class of special nonhomogeneous trees.Firstly,the devel-opment background and research status of the Markov chain field on the tree are introduced.Then the definition of this particular non-homogeneous tree is given.By constructing different auxiliary non-negative martingales independently,the Doob martingale convergence theorem is applied to al-most everywhere convergence condition.On the basis of previous studies on Markov chain,several concepts such as sample divergence and generalized gambling system are added.Extending it to m-ordered Markov chains on trees,we give and prove a class of strong deviation theorems for m-ordered non-homogeneous Markov chain on a special kind of non-homogeneous tree,a class of strong laws of large numbers for m-ordered non-homogeneous Markov chain on a special kind of non-homogeneous tree,and strong devi-ation properties of m-ordered non-homogeneous Markov chains on general-ized gambling systems in nonhomogeneous trees.Finally,for each theorem,the relevant conclusions derived from it are given and proved.