A Class of Local Limit Theorems for Markov Chains Field Indexed with the Generalized Bethe Tree on the Generalized Random Selection System

摘要

In this paper, we study the local limit theorems for the two-variate function sequence of Markov chain fields indexed by the the generalized Bethe tree with respect to the generalized random selection system by constructing a nonnegative martingale. As corollaries, a strong limit theorem for frequencies of states and ordered couples of states on Cayley tree is obtained. In the proof, we apply a new technique in the study of the Markov chain fields.