Some Researches On Strong Limit Theory For Nonsymmetric Markov Chain Indexed By Cayley Tree And Dependent Random Variables

The research of strong limit theory plays an important role in the history of probability theory. In recent years, the strong limit theory of random fields on trees and dependent random variables become the research Highlight of the scholars.Random fields on trees are applications on trees of theory of stochastic process—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 ofstate 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 be a researching emphases for many scholars. In recent years, Professor Yang Weiguo and his associates extend some strong limit theorems and Shannon-McMillan theorem from classical Markov chains to Markov chains on Bethe trees and Cayley trees. In the third chapter of this article , we first study the local convergence theorem for finite nonsymmetric Markov chain indexed by Cayley tree, then obtain some limit theorems for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain indexed by that tree, finally, we obtain the strong law of large numbers and asymptotic equipartition property (AEP) with a.e. convergence for finite nonsymmetric Markov chain indexed by that tree.Throughout the 1950s, the concept of dependent random variables was raised in some branches of the probability theory and mathematical statistics. The history and literature on the strong law of large numbers for dependent summands is not nearly as extensive and complete as it is for the case of independent summands. Lyons (1988) and Hu et al (2005, 2008) studied the convergence properties of sums of dependent random variables under second moment or covariance restrictions. In the fourth chapter of this article, by using the general" method of subsequence ", which was developed by Rajchman, we study the strong limit theory for dependent random variables by controlling the growth rates of p-th moment and covariance of them.