Strong Limit Theorems for Even-Odd Markov Chain Fields Indexed by Bethe Tree and Cayley Tree

In this article, we propose the definition of the even-odd Markov chain fields indexed by Bethe tree T_B,N (Cayley tree T_C,N). A strong limit theorem for the random variable sequence is proved by constructing two nonnegative martingales, and then a strong limit theorem for the even-odd Markov chain fields is obtained with application of the foregoing theorem. As its corollaries, we gain some strong limit theorems and coarse estimates for the frequencies of occurrence of states and ordered couples of states. Thus we generalize parts of strong limit theorems for Markov chain fields indexed by Bethe trees T_B,N and Cayley tree T_C,N and even-odd Markov chain fields indexed by Cayley tree T_C,2