Some limit properties for Markov chains indexed by a homogeneous tree

Abstract

We first study a local convergence theorem for countable Markov chains indexed by a homogeneous tree. As corollaries, we obtain some limit theorems for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a homogeneous tree. Finally, we obtain the strong law of large numbers and Shannon–McMillan theorem for finite Markov chains indexed by a homogeneous tree. In the proof, a new technique for establishing the strong limit theorems in probability theory is applied.

Introduction

Let T be a homogeneous tree on which each vertex has N+1 neighboring vertices. We first fix any vertex as the “root” and label it by 0. Let σ,τ be vertices of a tree. Write τ⩽σ if τ is on the unique path connecting 0 to σ, and |σ| for the number of edges on this path. For any two vertices σ,τ, denote by σ∧τ the vertex farthest from 0 satisfying

$$\text{σ∧τ⩽σ}\text{and}\text{σ∧τ⩽τ.}$$

If σ≠0, then we let $\text{σ}\text{̄}$ stand for the vertex satisfying $\text{σ}\text{⩽σ}$ and $\text{|}\text{σ}\text{|=|σ|−1}$ (we refer to σ as a son of $\text{σ}\text{̄}$). It is easy to see that the root has N+1 sons and all other vertices have N sons.

Definition Tree-indexed Markov chains, see Benjamini and Peres (1994)

Let T be a homogeneous tree, G={0,1,2,…} be countable state space, {X_σ,σ∈T} be a collection of G-valued random variables defined on the probability space $\text{(Ω,}\text{F}\text{,P)}$. Let

$$\text{p={p(x),x∈G}}$$

be a distribution on G, and

$$\text{P=(P(y|x)),}\text{x,y∈G}$$

be a stochastic matrix on G². If for any vertices σ,τ

$$\text{P(X}{}_{\mathrm{\sigma }}\text{=y|X}{}_{\mathrm{<mtext>\sigma </mtext>}}\text{=x}\text{and}\text{X}{}_{\mathrm{\tau }}\text{for}\text{τ∧σ⩽}\text{σ}\text{)=P(X}{}_{\mathrm{\sigma }}\text{=y|X}{}_{\mathrm{<mtext>\sigma </mtext>}}\text{=x)=P(y|x)}\text{∀x,y∈G}$$

and

$$\text{P(X}{}_0\text{=x)=p(x)}\text{∀x∈G,}$$

{X_σ,σ∈T} will be called G-valued Markov chains indexed by a homogeneous tree with the initial distribution (1) and transition matrix (2).

It is easy to see that a tree-indexed Markov chain is the particular case of a Markov random field on a tree. Two special finite tree-indexed Markov chains are introduced in Kemeny et al. (1976, p. 456), Spitzer (1975), and there the finite transition matrix is assumed to be positive and reversible to its stationary distribution, and this tree-indexed Markov chains ensure that the cylinder probabilities are independent of the direction we travel along a path. In this paper, we have no such assumption.

If |σ|=n, it is said to be on the nth level on a tree T. We denote by T⁽ⁿ⁾ the subtree of T containing the vertices from level 0 (the root) to level n, and L_n the set of all vertices on level n. Let B be a subgraph of T. Denote X^B={X_σ,σ∈B}, and denote by |B| the number of vertices of B. Let S(σ) be the set of all sons of vertices σ. It is easy to see that |S(0)|=N+1 and |S(σ)|=N, where σ≠0.

The subject of tree-indexed processes is rather young. Benjamini and Peres (1994) have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Berger and Ye (1990) have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Pemantle (1992) proved a mixing property and a weak law of large numbers for a PPG-invariant and ergodic random field on a homogeneous tree. Ye and Berger 1996, Ye and Berger 1998, by using Pemantle's result and a combinatorial approach, have studied the Shannon–McMillan theorem with convergence in probability for a PPG-invariant and ergodic random field on a homogeneous tree. Yang and Liu (2000) have studied a strong law of large numbers for the frequency of occurrence of states for Markov chains field on a homogeneous tree (a particular case of tree-indexed Markov chains and PPG-invariant random fields).

In this paper, we first study the local convergence theorem for countable Markov chains indexed by a homogeneous tree. As corollaries, we obtain some limit theorems for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by a homogeneous tree. Finally, we obtain the strong law of large numbers and Shannon–McMillan theorem with a.e. convergence for finite Markov chains indexed by a homogeneous tree. We use an approach of establishing a.e. convergence proposed in the works (see Liu and Yang 1996, Liu and Yang 2000) that differs from the traditional technique.