Some Limit Theorems For Markov Bichain

Probability theory is a branch of mathematics dealing with chance phenomena and has clearly discernible links with the real world. It is the framework foundations of many applying subjects, such as Information theory, Mathematics risk theory and Insurance theory for Actuaries etc. The strong limit theorems is one of the central questions for studying Ptobability theory.In this paper, a class of strong limit theorems for Markov bichain were studied by constructing a martingale and using the martingale convergence theorem. In the second chapter, first we discussed the properties of transition matrix for Markov bichain, then some limit theorems of occurrance of ordered couples for countable non-homogeneous Markov bichains were studied. Furthermore, Shannon-McMillan theorem was extended to the case of Markov bichain. In the third chapter, some limit theorems for functional of Markov bichain were obtained. In the fourth chapter, second oreder Markov chains as a special class of Markov bichain, was studied and a class of new strong limit theorems on the frequency of occurrence of m-tuple of states for arbitrary countable non-homogeneous two orders Markov chains were obtained. As corollaries, some strong limit theorems of equitable ratios of the frequency of occurrence of m-tuple of states for arbitrary countable non-homogeneous two orders Markov chains are obtained. In the last chapter, we summed up what we have done in this paper.