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

Abstract

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 n th 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 .