Some Stong Limit Theorems And Its Application To The Even-Odd Markov Chain Fields Indexed By A Extended Bethe Tree

In recent years,tree models have aroused extensive interests of physics , probability theory and information theory circles. The analytic technique which is first invented by professor Liu Wen is skillfully introduced to the study of strong limit theorems for Markov chain fields on trees by professor Liu Wen and his copartners. In this paper, the martingale technique is applied to continue the study in this respect.First of all, a class of strong limit theorems which extend the conclusions in [29] are proved by constructing two nonnegative martingales. Secondly,definitions of a extended Bethe tree,Bethe tree and Cayley tree are introduced.Further,the even-odd Markov chain fields and Markov chain fields indexed by a extended Bethe tree,Bethe tree and Cayley tree are defined.Subsequently the strong limit theorems proved above are applied to the study of all kinds of even-odd Markov chain fields and Markov chain fields defined above,which both extend the conclusions in [30] and [39] and prove a class of strong limit theorems for the even-odd Markov chain fields and Markov chain fields.As corollaries ,we obtain some strong limit theorems for the frequencies of occurrence of states and ordered couples of states for the even-odd Markov chain fields and Markov chain fields.Lastly,some coarse estimates are gained about the state occurrence frequencies for all kinds of even-odd Markov chain fields and Markov chain fields defined in the paper.