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A class of small deviation theorems for the random fields on an m rooted Cayley tree
Journal of Inequalities and Applications volume 2012, Article number: 1 (2012)
Abstract
In this paper, we are to establish a class of strong deviation theorems for the random fields relative to m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree.
2000 Mathematics Subject Classification: 60F15; 60J10.
1. Introduction
A tree is a graph G = {T, E} which is connected and contains no circuits. Given any two vertices σ, t(σ ≠ t ∈ T), let be the unique path connecting σ and t. Define the graph distance d (σ, t) to be the number of edges contained in the path .
Let T C,N be a Cayley tree. In this tree, the root (denoted by o) has only N neighbors and all other vertices have N + 1 neighbors. Let T B, N be a Bethe tree, on which each vertex has N + 1 neighboring vertices. Here both T C,N and T B,N are homogeneous tree. In this paper, we mainly consider an m rooted Cayley tree (see Figure 1). It is formed by a Cayley tree T C,N with the root o connecting with another vertex denoted by the the root -1, and then root -1 connecting with another vertex denoted by the root -2, and continuing to do the same work until the last vertex denoted by the root - (m - 1) is connected. When the context permits, this type of tree is denoted simply by T.
Let σ, t(σ, t ≠ o, -1, - 2,..., - (m - 1)) be vertices of an m rooted Cayler tree T. Write t ≤ σ if t is on the unique path connecting o to σ, and |σ | the number of edges on this path. For any two vertices σ, t(σ, t ≠ o, -1, - 2,..., - (m - 1)) of tree T, denote by σ ∧ t the vertex farthest from o satisfying σ ∧ t ≤ σ and σ ∧ t ≤ t.
The set of all vertices with distance n from the root o is called the n-th generation of T, which is denoted by L n . We say that L n is the set of all vertices on level n and especially root -1 is on the -1st level on tree T, root -2 is on the -2nd level. By analogy, root -(m - 1) is on the -(m - 1) th level. We denote by T(n)the subtree of an m rooted Cayley tree T containing the vertices from level -(m - 1) (the root -(m - 1)) to level n. Let t(t ≠ o, -1, -2, ..., -(m - 1)) be a vertex of an m rooted Cayley tree T. Predecessor of the vertex t is another vertex, which is nearest from t, on the unique path from root -(m - 1) to t. We denote the predecessor of t by 1 t , the predecessor of 1 t by 2 t and the predecessor of (n - 1) t by n t . We also say that n t is the n-th predecessor of t. XA = {X t , t ∈ A} is a stochastic process indexed by a set A, and denoted by |A| the number of vertices of A, xA is the realization of XA.
Let be a measure space, {X t , t∈T} be a collection of random variables defined on and taking values in G = {0,1,..., b - 1}, where b is a positive integer. Let P be a general probability distribution on . We will call P the random field on tree T. Denote the distribution of {X t , t ∈ T} under the probability measure P by
Let
f n (ω) is called entropy density of .
Let Q be another probability measure on the measurable space , and let the distribution of {X t , t ∈ T} under Q be
Let
h(P | Q) is called the sample divergence rate of P relative to Q.
Remark 1 If P = Q, h(P | Q) = 0 holds. By using the approach of Lemma 1 of Liu and Wang [1], we also can prove that h(P | Q) ≥ 0, P - a.e.; hence, h(P | Q) can be regarded as a measure of the Markov approximation of the arbitrary random field on T.
Definition 1 (see [2]) Let G = {0, 1,..., b - 1} and P(y|x1, x2,..., x m ) be a nonnegative functions on Gm+1. Let
If
then P is called an m-order transition matrix.
Definition 2 (see [2]). Let T be an m rooted Cayley tree, and let G = {0, 1,..., b - 1} be a finite state space, {X t , t ∈ T} be a collection of G-valued random variables defined on the probability space . Let Q be a probability on a measurable space .
Let
be a distribution on Gm, and
be m-order transition matrices. For any vertex t ∈ L n , n ≥ 1, if
and
then {X t , t ∈ T} is called a G-valued m th-order nonhomogeneous Markov chain indexed by an m rooted Cayley tree with the initial m dimensional distribution (5) and m-order transition matrices (6) under the probability measure Q, or called a T-indexed m th-order nonhomogeneous Markov chain under the probability measure Q.
We denote
and denote by and the realizations and , respectively.
Let {X t , t ∈ T} be an m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree T under the probability measure Q defined on above. It is easy to see that
In the following, we always assume that P(xT(n)), Q(xT(n)), q(x1,..., x m ), and {q n (y | x1,..., x m ), n ≥ 1} are all positive.
There have been some works on limit theorems for tree-indexed stochastic process. 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 [4] have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Pemantle [5] 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 [6, 7], 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 [8] have studied a strong law of large numbers for the frequency of occurrence of states for Markov chains field on a Bethe tree (a particular case of tree-indexed Markov chains field and PPG-invariant random field). Yang [9] has studied the strong law of large numbers for frequency of occurrence of state and Shannon-McMillan theorem for homogeneous Markov chains indexed by a homogeneous tree. Yang and Ye [10] have studied the strong law of large numbers and Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang [11] have studied the strong law of large numbers and Shannon-McMillan theorem for Markov chains indexed by an infinite tree with uniformly bounded degree. Recently, Shi and Yang [12] have also studied some limit properties of random transition probability for second-order nonhomogeneous Markov chains indexed by a tree. Peng et al. [13] have studied a class of strong deviation theorems for the random fields relative to homogeneous Markov chains indexed by a homogeneous tree. Shi and Yang [2] have studied the strong law of large numbers and Shannon-McMillan for the m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. Yang [14] has also studied a class of small deviation theorems for the sequences of N-valued random variables with respect to m th-order nonhomogeneous Markov chains.
In this paper, our main purpose is to extend Yang's [14] result to an m rooted Cayley tree. By introducing the sample divergence rate of any probability measure with respect to m th-order nonhomogeneous Markov measure on an m rooted Cayley tree, we establish a class of strong deviation theorems for the arbitrary random fields indexed by that tree with respect to m th-order nonhomogeneous Markov chains indexed by that tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree.
2. Main Results
Before giving the main results, we begin with a lemma.
Lemma 1 Let T be an m rooted Cayley tree, G = {0, 1,..., b - 1} be the finite state space. Let {X t , t ∈ T} be a collection of G-valued random variables defined on the measurable space . Let P and Q be two probability measures on the measurable space , and let {X t , t ∈ T} be an m th-order nonhomogeneous Markov chains indexed by tree T under probability measure Q. Let {g n (y1,..., y m +1), n ≥ 1} be a sequence of functions defined on Gm+1. Let . Set
and
where E Q denote the expectation under probability measure Q. Then is a nonnegative martingale under probability measure P.
Proof The proof is similar to Lemma 3 of Peng et al. [12], so the proof is omitted.
Theorem 1 Let T be an m rooted Cayley tree, {X t , t ∈ T} be a collection of random variables taking values in G = {0, 1,..., b - 1} defined on the measurable space . Let P and Q be two probability measures on the measurable space , such that {X t , t ∈ T} is an m th-order nonhomogeneous Markov chain indexed by T under Q. Let h(P | Q) be defined by (4), {g n (y1,..., y m +1), n ≥ 1} be a sequence of functions defined on Gm+1. Let c ≥ 0 be a constant. Set
Assume that there exists α > 0, such that ∀im ∈ Gm,
Let
where o < t < a. Thus, when 0 ≤ c ≤ t2A t , we have
In particular,
Proof Let t n (λ, ω) be defined by (11). By Lemma 1, is a non-negative martingale under probability measure P. By Doob's martingale convergence theorem, we have
Hence,
We have by (9), (10), (11) and (17)
By (4),(12) and (18)
This implies that
Let |λ| < t. By inequalities In x ≤ x -1(x > 0) and , and noticing that
We have
By (20) and (22), we have
When 0 < λ < t < α, we have by (23)
It is easy to see that when 0 < c < t2A t , the function f (λ) = λA t + c/λ attains, at , its smallest value . Letting in (24), we have
When c = 0, we have by (24)
Letting λ → 0+ in (26), we obtain
Hence, (25) also holds for c = 0. When -α < -t < λ < 0, by virtue of (23) it can be shown in a similar way that
Equation 15 follows from (25) and (28), Equation 15 implies (16) immediately. This completes the proof of the theorem. □
Theorem 2 Let
Let f n (ω) be defined by (2). Under the conditions of Theorem 1, when 0 ≤ c ≤ t2H t , we have
where H(p0,.... p b -1) denote the entropy of distribution (p0,..., p b -1), i.e.,
Proof In Theorem 1, let g k (y1,..., y m +1) = - In q k (y m +1 | y1,..., y m ) and α = 1, we have
Hence, ∀im ∈ Gm,
Noticing that
When 0 ≤ c ≤ t2H t , we have by (34),(29) and (15)
By (35), (9) and h(P|Q) ≥ 0,
By (36), (9) and (12), we have
This completes the proof of this theorem. □
Corollary 1 Under the conditions of Theorem 2, we have
If P << Q, then
In particular, if P = Q,
Proof Letting c = 0 in (30) and (31), Equation 39 follows. If P << Q, then h(P | Q) = 0, P - a.e.,(cf. see [15],P.121), i.e., P(D(0)) = 1. Hence, Equation 40 follows from (39). In particular, if P = Q, then h(P | Q) ≡ 0. Hence, (41) follows from (40). □
Theorem 3 Under the conditions of Theorem 1, if {g n (y1,.... y m +1), n ≥ 1} is uniformly bounded, i.e., there exists M > 0 such that |g n (y1,..., y m +1)| ≤ M, then when c ≥ 0, we have
Proof By (20) and (12) and the formula in line 2 of (22), we have
By the hypothesis of the theorem and the inequality ex - 1 - x ≤ |x|(e|x|- 1), we have
By (43) and (44)
When λ > 0, we have by (45)
Taking , and using the inequality
we have when c > 0
When λ < 0, it follows from (45) that
Taking in (49), and using (47), we have when c > 0
In a similar way, it can be shown that (48) and (50) also hold when c = 0. By (48) and (50), we have (42) holds. This completes the proof of this theorem.□
Corollary 2 Under the conditions of Theorem 1, let g(y1,..., y m +1) be any function defined on Gm+1. Let M = max g(y1,..., y m +1). Then when c ≥ 0,
Proof Letting g(y1,..., y m +1) = g n (y1,..., y m +1), n ≥ 1 in Theorem 3, this corollary follows.
In the following, let . Let be the number of (i1,..., i m ) in the collection of , that is
be the number of (i1,..., i m , i m +1) in the collection of , that is
Corollary 3 Let {X t , t ∈ T} be defined as before. Then for all i1,..., i m +1 ∈ G, c ≥ 0, we have
Proof Letting in Corollary 2.
and
Noticing that M = max g(y1,..., y m +1) = 1, , by (56) and (57) and Corollary 2, (54) holds. Similarly, we let (55) follows.
Corollary 4 Let {X t , t ∈ T} be defined as before.
If P = Q, then above equations hold Q - a.e..
Proof Letting c = 0 in Corollary 2 and Corollary 3, (58)-(60) follow from (51),(54) and (55). In particular, if P = Q, then h(P|Q) = 0, so (58)-(60) hold P - a.e., hence hold Q - a.e.
Definition 3 Let G = {0, 1,..., b - 1} be a finite state space and
be an m th-order transition matrix. Define a stochastic matrix as follows:
where
Then is called an m-dimensional stochastic matrix determined by the m th-order transition matrix.Q1.
Lemma 2 (see [16]). Let be an m-dimensional stochastic matrix determined by the m th-order transition matrix Q1. If the elements of Q1 are all positive, that is
then is ergodic.
Theorem 4 Let {X t , t ∈ T} be defined as Theorem 1. Let and f n (ω) defined by (52),(53) and (2), respectively. Let h(P|Q) and D(c) be defined by (4) and (12), respectively. Let the m th-order transition matrices defined by (6) be changeless with n, that is
or {X t , t ∈ T} is an m th-order homogeneous Markov chain indexed by tree T with the m th-order transition matrix Q1 under the probability measure Q. Let the m-dimensional stochastic matrix determined by Q1 be ergodic. Then for all i1,..., i m +1 ∈ G, we have
where {π(im), im ∈ Gm} is the stationary distribution determined by .
Proof Proof of Equation 66. Let km = (k1,..., k m ). If (65) holds, then we have by (63) and (52)
By (59) and (69), we have
Multiplying (70) by q(jm|im), adding them together for im ∈ Gm, and using (70) once again, we have
By induction, we have
where q(h)(jm|km) is the h th step probability determined by . We have by ergodicity
and . (66) follows from (71) and (72). By (66) and (60), Equation 67 follows easily.
Proof of Equation 68. By (66) and (53), we have