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In this paper, we first study the strong convergence theorem for
finite two-order nonhomogeneous Markov chains indexed by an two rooted Cayley tree, then we
obtain the strong law of large numbers for this Markov chains. Finally, we obtain the
Shannon-McMillan theorem with a.e. convergence for an two-order nonhomogeneous Markov chain
indexed by an two rooted Cayley tree.