A Class of Strong Limit Theorems for Nonhomogeneous Markov Chains Field Indexed by a Homogeneous Tree

In this paper, we study the strong limit theorems on the a.s. convergence for the harmonic mean of the transition probabilities of nonhomogeneous Markov chain indexed by the homogeneous tree. In the proof, we apply the tool of conditional moment generating functions and establishment of the consistent distribution functions to the study of strong limit theorems for Markov chains indexed by the homogeneous tree. As corollaries, some strong limit theorems for the Markov chains field on a homogeneous tree and the nonhomogeneous Markov chain are obtained. Some results which have been obtained are extended.