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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
Noticing that , by (39), (73) and (66), Equation 68 follows.□
3. Shannon-McMillan Theorem
Theorem 5 Let {X t , t ∈ T} be a G-valued m th-order nonhomogeneous Markov chain indexed by an m rooted Cayley tree under the probability measure Q with initial distribution (5) and m th-order transition matrices (6). Let and f n (ω) be defined as before. Let
be another positive m th-order transition matrix. Let be an m dimension transition matrix determined by Q1. If
then
where {π(im), im ∈ Gm} is the stationary distribution determined by . In particular, if P = Q, then above equations hold Q - a.e.
Proof By (59), (75), (52) and (66), (76) follows immediately. Similarly, by (60), (75), and (53), (77) follows. It follows from (75) and Cesaro average that
Notice that
By (79), (80), (39), (73) and (66), (78) follows. In particular, if P = Q, then h(P|Q) = 0. (76), (77) and (78) also holds P - a.e. □
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Acknowledgements
The authors would like to thank the referees for their valuable suggestions and comments. This work is supported by the Research Foundation for Advanced Talents of Jiangsu University (11JDG116) and the National Natural Science Foundation of China (11071104,11171135,71073072).
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ZS, WY and LT carried out the design of the study and performed the analysis. WP participated in its design and coordination. All authors read and approved the final manuscript.
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Shi, Z., Yang, W., Tian, L. et al. A class of small deviation theorems for the random fields on an m rooted Cayley tree. J Inequal Appl 2012, 1 (2012). https://doi.org/10.1186/1029-242X-2012-1
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DOI: https://doi.org/10.1186/1029-242X-2012-1