STRONG LAW OF LARGE NUMBERS FOR MARKOV CHAINS INDEXED BY SPHERICALLY SYMMETRIC TREES

In this paper, we main consider spherically symmetric tree T. First, under the condition lim sup(n ->infinity) vertical bar T-(n)vertical bar/vertical bar L-n vertical bar < infinity, 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,T-N and Bethe tree T-B,T-N.