In this paper, we are to establish a class of strong deviation theorems for the random fields relative to m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree. 2000 Mathematics Subject Classification: 60F15; 60J10.

In this paper, we main consider spherically symmetric tree T. First, under the condition lim sup_n→∞ |T⁽ⁿ⁾|/|L_n|<∞, we investigate the strong law of large numbers (SLLNs) for T-indexed Markov chains on the nth level of T. Then, combining the Stolz theorem, we obtain the SLLNs on T. Finally, we get Shannon–McMillan theorem for T-indexed Markov chains. The obtained theorems are generalizations of some known results on Cayley tree T_{C, N} and Bethe tree T_{B, N}.