Abstract: In this paper, we study the generalized entropy ergodic theorem for Markov chains indexed by a homogeneous tree. The entropy ergodic theorem studies the asymptotic equipartition property of information source in the information theory, and the theory of stochastic processes indexed by tree has become one of the research branches in probability theory recently. We first introduce the definition of the generalized entropy density. Then we prove the strong limit theorem of certain random variables by constructing a single parameter class of random variables with means 1 and using the Markov inequality and the Borel-Cantelli lemma. Finally, from the corollaries of the above theorem, we obtain the strong law of large numbers for the delayed average of the number of occurrences of some state and the generalized entropy ergodic theorem for finite Markov chains indexed by a Cayley tree, which generalize some known results.

Key words: Cayley tree, Markov chains, strong law of large numbers, generalized entropy ergodic theorem