Bethe树和Cayley树上奇偶马尔可夫链场的强极限定理

本文介绍了N元Bethe树TB,N(N元Cayley树TC,N)上的奇偶马尔可夫链场的定义,并通过构造两个非负鞅证得了随机变量序列的强极限定理,应用此强极限定理获得了奇偶马尔可夫链场上的一个强极限定理,作为它的推论得到了状态和状态序偶出现频率的一类强极限定理及其估计,从而推广了关于N元Bethe树上马氏链场和二进树上奇偶马氏链场的部分强极限定理.

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 $T_{C,2}$