A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system

Abstract

We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states, ordered couples of states, and the conditional expectation of a bivariate function on Cayley tree.