The Generalized Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree.

Abstract

In this paper, we study the generalized entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree. Firstly, by constructing a class of random variables with a parameter and the mean value of one, we establish a strong limit theorem for delayed sums of the bivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we prove the strong law of large numbers for the frequencies of occurrence of states of delayed sums and the generalized entropy ergodic theorem. As corollaries, we generalize some known results.