Some Limit Properties of the Harmonic Mean of Transition Probabilities for Markov Chains in Markovian Environments Indexed by Cayley’s Trees

A tree T is a graph which is connected and doesn’t contain any circuits. Given any two vertices α ̸ = β ∈ T, let αβ be the unique path connecting α and β. Define the graph distance d(α, β) to be the number of edges contained in the path αβ. Let T be an infinite tree with root 0.The set of all vertices with distance n from the root is called the nth generation of T, which is denoted by L n . We denote by T(n) the union of the first n generations of T. For each vertex t, there is a unique path from 0 to t and |t| for the number of edges on this path. We denote the first predecessor of t by 1t. The degree of a vertex is defined to be the number of neighbors of it. If every vertex of the tree has degree d + 1, we say it is Cayley’s tree, which is denoted by T C,d . Thus, the root vertex has d + 1 neighbors in the first generation and every other vertex has d neighbors in the next generation. For any two vertices s and t of tree T, write s ≤ t if s is on the unique path from the root 0 to t. We denote by s ∧ t the farthest vertex from 0 satisfying s ∧ t ≤ s and s ∧ t ≤ t. We use the notation XA = {X t , t ∈ A} and denote by |A| the number of vertices of A. In the following, we always let T denote the Cayley tree T C,d . A tree-indexed Markov chain is the particular case of a Markov random field on a tree. Kemeny et al. [1] and Spitzer [2] introduced two special finite tree-indexed Markov chains with finite transition matrix which is assumed to be positive and reversible to its stationary distribution, and these treeindexed Markov chains ensure that the cylinder probabilities are independent of the directionwe travel along a path. In this paper, we omit such assumption and adopt another version of the definition of tree-indexed Markov chains which is put forward by Benjamini and Peres [3]. Yang and Ye[4] extended it to the case of nonhomogeneous Markov chains indexed by infinite Cayley’s tree and we restate it here as follows.