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

Huilin Huang

doi:10.1155/2013/961571

Abstract

We prove some limit properties of the harmonic mean of a random transition probability for finite Markov chains indexed by a homogeneous tree in a nonhomogeneous Markovian environment with finite state space. In particular, we extend the method to study the tree-indexed processes in deterministic environments to the case of random enviroments.

Highlights

A tree T is a graph which is connected and doesn’t contain any circuits
Define the graph distance d(α, β) to be the number of edges contained in the path αβ
The set of all vertices with distance n from the root is called the nth generation of T, which is denoted by Ln

Summary

Introduction

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. 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. {Xt, t ∈ T} will be called X-valued nonhomogeneous Markov chains indexed by infinite Cayley’s tree with initial distribution (1) and transition probability matrices {Pt, t ∈ T}.

Results

Conclusion