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

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.

1. Introduction

A tree𝑇is a graph which is connected and doesn't contain any circuits. Given any two vertices𝛼≠𝛽∈𝑇, let𝛼𝛽be the unique path connecting𝛼and𝛽. Define the graph distance𝑑(𝛼,𝛽)to be the number of edges contained in the path𝛼𝛽.

Let𝑇be an infinite tree with root0. The set of all vertices with distance𝑛from the root is called the𝑛th generation of𝑇, which is denoted by𝐿_𝑛. We denote by𝑇^(𝑛)the union of the first𝑛generations of𝑇. For each vertex𝑡, there is a unique path from0to𝑡 and|𝑡|for the number of edges on this path. We denote the first predecessor of𝑡by 1𝑡. The degree of a vertex is defined to be the number of neighbors of it. If every vertex of the tree has degree𝑑+1, we say it is Cayley’s tree, which is denoted by𝑇_𝐶,𝑑. Thus, the root vertex has𝑑+1neighbors in the first generation and every other vertex has𝑑neighbors in the next generation. For any two vertices𝑠 and 𝑡of tree𝑇, write𝑠≤𝑡if𝑠is on the unique path from the root0to𝑡. We denote by𝑠∧𝑡the farthest vertex from0satisfying𝑠∧𝑡≤𝑠and𝑠∧𝑡≤𝑡. We use the notation𝑋^𝐴={𝑋_𝑡,𝑡∈𝐴}and denote by|𝐴|the number of vertices of𝐴.

In the following, we always let𝑇denote the Cayley tree𝑇_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 tree-indexed Markov chains ensure that the cylinder probabilities are independent of the direction we 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.

Definition 1 (T-indexed nonhomogeneous Markov chains (see [4])). Let𝑇be an infinite Cayley tree, 𝒳a finite state space, and{𝑋_𝑡,𝑡∈𝑇}a stochastic process defined on probability space(Ω,𝐹,𝐏), which takes values in the finite set𝒳. Let 𝑝=𝑝(𝑖),𝑖∈𝒳 (1) be a distribution on𝒳 and 𝑃_𝑡=𝑃_𝑡𝑗∣𝑖,𝑖,𝑗∈𝒳, (2) a transition probability matrix on𝒳². If, for any vertex𝑡, 𝐏𝑋_𝑡=𝑗∣𝑋_1𝑡=𝑖,𝑋_𝑠=𝑥_𝑠𝑋for𝑡∧𝑠≤1𝑡=𝐏_𝑡=𝑗∣𝑋_1𝑡=𝑖=𝑃_𝑡𝐏𝑋𝑗∣𝑖,∀𝑖,𝑗∈𝒳,₀=𝑖=𝑝(𝑖)∀𝑖∈𝒳, (3)(4) then{𝑋_𝑡,𝑡∈𝑇}will be called 𝒳-valued nonhomogeneous Markov chains indexed by infinite Cayley’s tree with initial distribution (1) and transition probability matrices{𝑃_𝑡,𝑡∈𝑇}.

The subject of tree-indexed processes has been deeply studied and made abundant achievements. Benjamini and Peres [3] have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Berger and Ye [5] have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Ye and Berger [6, 7], by using Pemantle's result [8] 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 [9] and Yang [10] have studied a strong law of large numbers for Markov chains fields on a homogeneous tree (a particular case of tree-indexed Markov chains and PPG-invariant random fields). Yang and Ye [4] have established the Shannon-McMillan theorem for nonhomogeneous Markov chains on a homogeneous tree. Huang and Yang [11] has studied the strong law of large numbers for finite homogeneous Markov chains indexed by a uniformly bounded infinite tree.

The previous results are all about tree-indexed processes in deterministic environments. Recently, we are interested in random fields indexed by trees in random environments. In the rest of this paper we formulate a model of Markov chain indexed by trees in random environment especially in Markovian environment and study some limit properties of the harmonic mean of random transition probability for finite Markov chains indexed by a homogeneous tree in a nonhomogeneous Markovian environment. We are also interested in the strong law of large numbers of Markov chains indexed by trees in random environments which we will prepare for in another paper.

Definition 2. Let𝑇be an infinite Cayley tree,𝒳andΘtwo finite state spaces. Suppose that{𝜉_𝑡,𝑡∈𝑇}is aΘ-valued random field indexed by𝑇, if, for any vertex𝑡∈𝑇, 𝐏𝑋_𝑡=𝑗∣𝑋_1𝑡=𝑖,𝑋_𝑠=𝑥_𝑠,for𝑡∧𝑠≤1𝑡;𝜉_𝑟,𝑟∈𝑇=𝑃_𝜉1𝑡𝐏𝑋𝑖,𝑗,a.s.,₀=𝑖∣𝜉_𝑟,𝑟∈𝑇=𝑃𝑖∣𝜉₀∀𝑖∈𝒳 (5)(6) for each𝑖,𝑗∈𝒳, where 𝑃_𝑥=(𝑃_𝑥(𝑖,𝑗))_{𝑖,𝑗∈𝒳}, 𝑥∈Θis a family of stochastic matrices. Then we call{𝑋_𝑡,𝑡∈𝑇}a Markov chain indexed by tree𝑇in a random environment{𝜉_𝑡,𝑡∈𝑇}. The𝜉^′_𝑡𝑠are called the environmental process or control process indexed by tree 𝑇. Moreover, if{𝜉_𝑡,𝑡∈𝑇}is a T-indexed Markov chain with initial distribution𝜇={𝜇(𝜃),𝜃∈Θ}and one-step transition probability matrices𝐾_𝑡={𝐾_𝑡(𝛼,𝛽),𝛼,𝛽∈Θ,𝑡∈𝑇}, we call{𝑋_𝑡,𝑡∈𝑇}a Markov chain indexed by tree𝑇in a nonhomogeneous Markovian environment.

Remark 3. If every vertex has degree two, then our model of Markov chain indexed by homogeneous tree with Markovian environments is reduced to the model of Markov chain in Markovian environments which was introduced by Cogburn [12] in spirit.

Remark 4. We also point out that our model is different from the tree-valued random walk in random environment (RWRE) that is studied by Pemantle and Peres [13] and Hu and Shi [14]. For the model of RWRE on the trees that they studied, the random environment process𝜔∶=(𝜔(𝑥,⋅),𝑥,𝑦∈𝑇)is a family of i.i.d nondegenerate random vectors and the process𝑋={𝑋_𝑛,𝑛∈𝐙⁺}is a nearest-neighbor walk satisfying some conditions. But in our model,our environmental process𝜉={𝜉_𝑡,𝑡∈𝑇}can be any Markov chain indexed by trees. Given𝜉, the process{𝑋_𝑡,𝑡∈𝑇}is another Markov chain indexed by trees with the law𝑃_𝜉.

In this paper we assume that{𝜉_𝑡,𝑡∈𝑇}is a nonhomogeneous𝑇-indexed Markov chain on state spaceΘ. The probability of going from𝑖to𝑗in one step in the𝜃th environment is denoted by𝑃_𝜃(𝑖,𝑗). We also suppose that the one-step transition probability of going from𝛼to𝛽for nonhomogeneous𝑇-indexed Markov chain{𝜉_𝑡,𝑡∈𝑇}is𝐾_𝑡(𝛼,𝛽). In this case,{𝜉_𝑡,𝑋_𝑡,𝑡∈𝑇}is a Markov chain indexed by𝑇with initial distribution𝑞=(𝑞(𝜃,𝑖))and one-step transition onΘ×𝒳 determined by 𝑃_𝑡𝛼,𝑖;𝛽,𝑗=𝐾_𝑡𝛼,𝛽𝑃_𝛼𝑖,𝑗, (7) where𝑞(𝜃,𝑖)=𝐏(𝜉₀=𝜃,𝑋₀=𝑖). Then{𝜉_𝑡,𝑋_𝑡,𝑡∈𝑇}will be called the bichain indexed by tree𝑇. Obviously, we have 𝐏𝜉^𝑇(𝑛)=𝛼^𝑇(𝑛),𝑋^𝑇(𝑛)=𝑥^𝑇(𝑛)=𝑞𝛼₀,𝑥_{0𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝛼_1𝑡,𝑥_1𝑡;𝛼_𝑡,𝑥_𝑡. (8)

2. Main Results

For every finite𝑛∈𝐍, let{𝑋_𝑡,𝑡∈𝑇}be a Markov chain indexed by an infinite Cayley tree𝑇in Markovian environment{𝜉_𝑡,𝑡∈𝑇}, which is defined as in Definition 2. Now we suppose that𝑔_𝑡(𝛼,𝑖,𝛽,𝑗)are functions defined onΘ×𝒳×Θ×𝒳. Let𝜆be a real number, 𝐿₀={0},ℱ_𝑛=𝜎(𝜉^𝑇(𝑛),𝑋^𝑇(𝑛)); now we define a stochastic sequence as follows: 𝜑_𝑛()=𝑒𝜆,𝜔^{𝜆∑𝑡∈𝑇(𝑛)⧵{0}𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∏_{𝑡∈𝑇(𝑛)⧵{0}}𝐸𝑒^{𝜆𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∣𝜉_1𝑡,𝑋_1𝑡. (9) At first we come to prove the following fact.

Lemma 5. {𝜑_𝑛(𝜆,𝜔),ℱ_𝑛,𝑛≥1}is a nonnegative martingale.

Proof of Lemma 5. Obviously, we have 𝐏𝜉^𝐿𝑛=𝛼^𝐿𝑛,𝑋^𝐿𝑛=𝑥^𝐿𝑛∣𝜉^{𝑇(𝑛−1)}=𝛼^{𝑇(𝑛−1)},𝑋^{𝑇(𝑛−1)}=𝑥^{𝑇(𝑛−1)}=𝐏𝜉^𝑇(𝑛)=𝛼^𝑇(𝑛),𝑋^𝑇(𝑛)=𝑥^𝑇(𝑛)𝐏𝜉^{𝑇(𝑛−1)}=𝛼^{𝑇(𝑛−1)},𝑋^{𝑇(𝑛−1)}=𝑥^{𝑇(𝑛−1)}=_{𝑡∈𝐿𝑛}𝐏𝜉_𝑡=𝛼_𝑡,𝑋_𝑡=𝑥_𝑡∣𝜉_1𝑡=𝛼_1𝑡,𝑋_1𝑡=𝑥_1𝑡. (10) Here, the second equation holds because of the fact that{𝜉_𝑡,𝑋_𝑡,𝑡∈𝑇}is a bichain indexed by tree𝑇and (8) is being used. Furthermore, we have 𝐸𝑒^{𝜆∑𝑛𝑡∈𝐿𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∣ℱ_𝑛−1=_{𝛼𝐿𝑛,𝑥𝐿𝑛}𝑒^{𝜆∑𝑛𝑡∈𝐿𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝛼𝑡,𝑥𝑡)}×𝐏𝜉^𝐿𝑛=𝛼^𝐿𝑛,𝑋^𝐿𝑛=𝑥^𝐿𝑛∣𝜉^{𝑇(𝑛−1)},𝑋^{𝑇(𝑛−1)}=_{𝛼𝐿𝑛,𝑥𝐿𝑛𝑡∈𝐿𝑛}𝑒^{𝜆𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝛼𝑡,𝑥𝑡)}×𝐏𝜉_𝑡=𝛼_𝑡,𝑋_𝑡=𝑥_𝑡∣𝜉_1𝑡,𝑋_1𝑡=_{𝑡∈𝐿𝑛𝛼𝑡,𝑥𝑡∈Θ×𝒳}𝑒^{𝜆𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝛼𝑡,𝑥𝑡)}×𝐏𝜉_𝑡=𝛼_𝑡,𝑋_𝑡=𝑥_𝑡∣𝜉_1𝑡,𝑋_1𝑡=_{𝑡∈𝐿𝑛}𝐸𝑒^{𝜆𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∣𝜉_1𝑡,𝑋_1𝑡a.s. (11) On the other hand, we also have 𝜑_𝑛(𝜆,𝜔)=𝜑_𝑛−1()×𝑒𝜆,𝜔_{𝜆∑𝑛𝑡∈𝐿𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∏_{𝑡∈𝐿𝑛}𝐸𝑒^{𝜆𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∣𝜉_1𝑡,𝑋_1𝑡. (12) Combining (11) and (12), we get 𝐸𝜑_𝑛(𝜆,𝜔)∣ℱ_𝑛−1=𝜑_𝑛−1(𝜆,𝜔)a.s. (13) Thus, we complete the proof of Lemma 5.

Theorem 6. Let{𝑋_𝑡,𝑡∈𝑇}be a Markov chain indexed by an infinite Cayley tree𝑇in a nonhomogeneous Markovian environment{𝜉_𝑡,𝑡∈𝑇}. Suppose that the initial distribution and the transition probability functions satisfy 𝑞𝛼₀,𝑥₀>0,𝑃_𝑡𝛼,𝑖;𝜃,𝑗>0,𝑓𝑜𝑟∀𝛼₀,𝛼,𝜃∈Θ;𝑥₀𝑎,𝑖,𝑗∈𝒳,_𝑡=min𝑃_𝑡𝛼,𝑖;𝜃,𝑗,𝛼,𝜃∈Θ;𝑖,𝑗∈𝒳,𝑡∈𝑇⧵{0}, (14) if there exist two positive constants𝑐and𝑚such that limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}𝑒^{𝑐/𝑎𝑡}=𝑚<∞. (15) Denote|Θ|=𝑀, |𝒳|=𝑁, and 𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡=𝑃_𝑡𝜉_𝑡,𝑋_𝑡∣𝜉_1𝑡,𝑋_1𝑡; (16) then we have lim_𝑛→∞𝑇^(𝑛)∑_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡⁻¹=1𝑀𝑁𝑎.𝑠. (17)

Proof. By Lemma 5, we have known that{𝜑_𝑛(𝜆,𝜔),ℱ_𝑛,𝑛≥1}is a nonnegative martingale. According to Doob martingale convergence theorem, we have lim_𝑛𝜑_𝑛(𝜆,𝜔)=𝜑(𝜆,𝜔)<∞a.s., (18) so that lim_𝑛→∞ln𝜑_𝑛(𝜆,𝜔)𝑇^(𝑛)=0a.s., (19) which implies that limsup_𝑛→∞ln𝜑_𝑛(𝜆,𝜔)𝑇^(𝑛)≤0a.s. (20) We arrive at limsup_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}}𝜆𝑔_𝑡𝜉_1𝑡,𝑋_1𝑡,𝜉_𝑡,𝑋_𝑡−ln𝐸𝑒^{𝜆𝑔𝑡(𝜉1𝑡,𝑋1𝑡,𝜉𝑡,𝑋𝑡)}∣𝜉_1𝑡,𝑋_1𝑡≤0a.s. (21) Combining (20) with the inequalitiesln𝑥≤𝑥−1(𝑥>0)and0≤𝑒^𝑥−1−𝑥≤2⁻¹𝑥²𝑒^|𝑥| and taking𝑔_𝑡(𝜉_1𝑡,𝑋_1𝑡,𝜉_𝑡,𝑋_𝑡)=𝑃_𝑡(𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡)⁻¹, it follows that limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡⁻¹𝜆−𝜆𝑀𝑁≤limsup_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}}ln𝐸𝑒^{𝜆𝑃𝑡(𝜉1𝑡,𝑋1𝑡;𝜉𝑡,𝑋𝑡)−1}∣𝜉_1𝑡,𝑋_1𝑡−𝜆𝑀𝑁≤limsup_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}}𝐸𝑒^{𝜆𝑃𝑡(𝜉1𝑡,𝑋1𝑡;𝜉𝑡,𝑋𝑡)−1}∣𝜉_1𝑡,𝑋_1𝑡−1−𝜆𝑀𝑁≤limsup_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}𝛼∈Θ𝑖∈𝒳}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡×𝑒;𝛼,𝑖_{𝜆/𝑃𝑡(𝜉1𝑡,𝑋1𝑡;𝛼,𝑖)}𝜆−1−𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡≤𝜆;𝛼,𝑖₂2limsup_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}𝛼∈Θ𝑖∈𝒳}1𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡𝑒;𝛼,𝑖^{|𝜆|/𝑃𝑡(𝜉1𝑡,𝑋1𝑡;𝛼,𝑖)}≤𝜆²𝑀𝑁2limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}1𝑎_𝑡𝑒^{|𝜆|/𝑎𝑡}a.s. (22) Note that the following elementary fact holds: max_𝑥>0𝑥𝛾^𝑥𝑒=−⁻¹ln𝛾,0<𝛾<1. (23) Let0<𝜆<𝑐. It follows from (15), (22), and (23) that limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡⁻¹≤−𝑀𝑁𝜆𝑀𝑁2limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}1𝑎_𝑡𝑒^{𝜆/𝑎𝑡}=𝜆𝑀𝑁2limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}1𝑎_𝑡𝑒^{𝜆−𝑐1/𝑎𝑡}𝑒^{𝑐/𝑎𝑡}≤𝜆𝑀𝑁2𝑒(𝑐−𝜆)𝑚a.s. (24) Here, the second equation holds because by letting𝜆→0⁺in inequality (24), we get limsup_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡⁻¹−𝑀𝑁≤0a.s. (25) If−𝑐<𝜆<0, similar to the analysis of inequality (24), by using (15), (22), and (23) again, we can arrive at liminf_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡⁻¹≥−𝑀𝑁𝜆𝑀𝑁2limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}1𝑎_𝑡𝑒^{−𝜆/𝑎𝑡}≥𝜆𝑀𝑁2limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}1𝑎_𝑡𝑒^−𝜆𝑒^{𝑐1/𝑎𝑡}𝑒^{𝑐/𝑎𝑡}≥𝜆𝑀𝑁2𝑒(𝜆+𝑐)𝑚a.s. (26) Letting𝜆→0⁻in inequality (26), we get liminf_𝑛→∞1𝑇^(𝑛)×_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝜉_1𝑡,𝑋_1𝑡;𝜉_𝑡,𝑋_𝑡⁻¹−𝑀𝑁≥0a.s. (27) Combining (25) and (27), we obtain that our assertion (17) is true.

Corollary 7. Let{𝑋_𝑡,𝑡∈𝑇}be a nonhomogeneous Markov chain indexed by an infinite Cayley tree𝑇. Suppose that the initial distribution and the transition probability functions satisfy 𝑝𝑥₀>0,𝑃_𝑡𝑎𝑖,𝑗>0,_𝑡=min𝑃_𝑡𝑖,𝑗,𝑖,𝑗∈𝒳,𝑡∈𝑇⧵{0}, (28) if there exist two positive constants𝑐and𝑚such that limsup_𝑛→∞1𝑇^(𝑛)_{𝑡∈𝑇(𝑛)⧵{0}}𝑒^{𝑐/𝑎𝑡}=𝑚<∞. (29) Denote|𝒳|=𝑁; then we have lim_𝑛→∞𝑇^(𝑛)∑_{𝑡∈𝑇(𝑛)⧵{0}}𝑃_𝑡𝑋_𝑡∣𝑋_1𝑡⁻¹=1𝑁𝑎.𝑠. (30)

Proof. If we takeΘ={𝜃}, that is,|Θ|=1, then the model of Markov chain indexed by tree𝑇in Markovian environment reduces to the formulation of a nonhomogeneous Markov chain indexed by tree𝑇. Then we arrive at our conclusion (30) directly from Theorem 6.

Corollary 8 (see [15]). Let{𝑋_𝑛,𝑛≥0}be a Markov chain in a nonhomogeneous Markovian environment{𝜉_𝑛,𝑛≥0}. Suppose that the initial distribution and the transition probability functions satisfy 𝑞𝛼₀,𝑥₀>0,𝑃_𝑛𝑎𝛼,𝑖;𝜃,𝑗>0,_𝑛=min𝑃_𝑛𝛼,𝑖;𝜃,𝑗,𝛼,𝜃∈Θ;𝑖,𝑗∈𝒳,𝑛≥1, (31) if there exist two positive constants𝑐and𝑚such that limsup_𝑛→∞1𝑛^𝑛_𝑘=1𝑒^{𝑐/𝑎𝑘}=𝑚<∞. (32) Denote|Θ|=𝑀, |𝒳|=𝑁, and 𝑃_𝑘𝜉_𝑘−1,𝑋_𝑘−1;𝜉_𝑘,𝑋_𝑘=𝑃_𝑘𝜉_𝑘,𝑋_𝑘∣𝜉_𝑘−1,𝑋_𝑘−1; (33) then we have lim_𝑛→∞𝑛∑^𝑛_𝑘=1𝑃_𝑘𝜉_𝑘−1,𝑋_𝑘−1;𝜉_𝑘,𝑋_𝑘⁻¹=1𝑀𝑁𝑎.𝑠. (34)

Proof. If every vertex of the tree𝑇has degree2, then the nonhomogeneous Markov chain indexed by tree𝑇degenerates into the nonhomogeneous Markov chain on line; thus, this corollary can be obtained from Theorem 6 directly.

Acknowledgment

This work was supported by the National Natural Science Foundation of China no. 11201344. The author declares that there is no conflict of interests regarding the publication of this paper.