Some limit properties for Markov chains indexed by a homogeneous tree.
The author studies tree-indexed Markov chains (TIMC) which are the particular case of a Markov random field on a tree. TIMC were introduced by \textit{I. Benjamini} and \textit{Y. Peres} [Ann. Probab. 22, 219--243 (1994; Zbl 0793.60080)], who investigated the recurrence and ray-recur\-rence for them. Many aspects of TIMC have been studied before, for example, the existence of entropy rate for some stationary random fields on a homogeneous tree, a mixing property, a weak law of large numbers, the Shannon-McMillan theorem with convergence in probability for a PPG-invariant and ergodic random field on a homogeneous tree. In this paper, the local convergence theorem for countable Markov chains indexed by a homogeneous tree is studied. Also, the strong law of large numbers and the Shannon-McMillan theorem with a.e. convergence for finite Markov chains indexed by a homogeneous tree are obtained.
scientific article; zbMATH DE number 2041555