A Type of Shannon-McMillan Approximation Theorems for Second-Order Nonhomogeneous Markov Chains Indexed by a Double Rooted Tree

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1 Caspian Journal of Applied Mathematics, Ecology and Economics V. 2, No, 204, July ISSN A Type of Shannon-McMillan Approximation Theorems for Second-Order Nonhomogeneous Markov Chains Indexed by a Double Rooted Tree Wang Kangkang, Zong Decai Abstract. In this paper, a class of small deviation theorems for the relative entropy densities of arbitrary random field on a double rooted tree are discussed by comparing between the arbitrary measure µ and the second-order nonhomogeneous Markov measure µ Q on the double rooted tree. As corollaries, some Shannon-McMillan theorems for the arbitrary random field, secondorder Markov chain field and a limit property for the random conditional entropy of second-order homogeneous Markov chain on the double rooted tree are obtained. The existing result is extended. Key Words and Phrases: Shannon-McMillan theorem, a double rooted tree, arbitrary random field, second-order nonhomogeneous Markov chain, relative entropy density Mathematics Subject Classifications: 60F5. Introduction A tree is a graph S = {T,E} which is connected and contains no circuits. Given any two vertices σ,t( σ t T), let σt 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 σt. Let T o be an arbitrary infinite tree that is partially finite (i.e. it has infinite vertices, and each vertex connects with finite vertices) and has a root o. Meanwhile, we consider another kind of double root tree T, that is, it is formed with the root o of T o connecting with an arbitrary point denoted by the root. For a better explanation of the double root tree T, we take Cayley tree T C,N for example. It s a special case of the tree T o, the root o of Cayley tree has N neighbors and all the other vertices of it have N + neighbors each. The double root tree T C,N (see Fig.) is formed with root o of tree T C,N connecting with another root. Let σ, t be vertices of the double root tree T. Write t σ (σ,t o, ) if t is on the unique path connecting o to σ, and σ for the number of edges on this path. For any Corresponding author. 9 c 203 CJAMEE All rights reserved.

2 20 Wang Kangkang, Zong Decai two vertices σ, t (σ,t o, ) of the 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 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 is on the st level on tree T. We denote by T (n) the subtree of the tree T the subtree of the tree T o containing the vertices from level 0 (the root o) to level n. Let t( o, ) be a vertex of the tree T. We denote the first predecessor of t by t, the second predecessor of t by 2 t, and denote by n t the n-th predecessor of t. Let X A = {X t,t A}, and let x A be a realization of X A and denote by A the number of vertices of A. containing the vertices from level (the root ) to level n and denote by T (n) o level 3 level 2 level level 0 level- t t 2 t root o root- Fig. Double root tree T C,2 Definition Let S = {s,s 2,,s M } and Q(z y,x) be a nonnegative function on S 3. Let Q = ((Q(z y,x)), Q(z y,x) 0, x,y,z S. If Q(z y,x) =, z S then Q is called a second-order transition matrix. Definition 2 Let T be a double root tree and S = {s,s 2,,s M } be a finite state space, and {X t,t T} be a collection of S-valued random variables defined on the probability space (Ω, F, Q). Let be a distribution on S 2, and q = (q(x,y)), x,y S () Q t = (Q t (z y,x)), x,y,z S, t T\{o}{ } (2)

3 2 be a collection of second-order transition matrices. For any vertex t (t o, ), if and Q(X t = z X t = y,x 2t = x,and X σ for σ t 2 t ) = Q(X t = z X t = y,x 2t = x) = Q t (z y,x) x,y,z S (3) Q(X = x,x o = y) = q(x,y), x,y S, (4) then {X t,t T} is called a S-valued second-order nonhomogeneous Markov chain indexed by a tree T with the initial distribution () and second-order transition matrices (2), or called a T-indexed second-order nonhomogeneous Markov chain. Definition 3. Let (Q t = Q t (z x,y),t T (n) \{o, }) and q = (q(x,y)) be defined as before, µ Q be a second-order nonhomogeneous Markov measure on (Ω,F). If µ Q (x T(n) ) = q(x 0,x ) µ Q (x 0,x ) = q(x 0,x ) (5) Q t (x t x t,x 2t ), n, (6) then µ Q will be called a second-order Markov chains field on an infinite tree T determined by the stochastic matrices Q t and the initial distribution q. Let µ be an arbitrary probability measure defined on (Ω,F), log is the natural logarithmic. Denote f n (ω) = logµ(xt(n) T (n) ). (7) f n (ω) is called the entropy density on subgraph T (n) with respect to the measure µ. If µ = µ Q, then by (6), (7) we get f n (ω) = [logq(x 0,X )+ Q t (X t X t,x 2t )]. (8) The convergence of f n (ω) in a sense (L convergence, convergence in probability, or almost sure convergence) is called Shannon-McMillan theorem or the asymptotic equipartition property(aep) in information theory. There have been some works on limit theorems for tree-indexed stochastic processes. Benjamini and Peres [] have given the notion of the tree-indexed Markov chains and studied the recurrence and ray-recurrence for them. Berger and Ye [2] have studied the existence of entropy rate for some stationary random fields on a homogeneous tree. Ye and Berger (see [4],[5] ), by using Pemantle s result [3] and a combinatorial approach, have studied the Shannon-McMillan theorem with convergence in probability for a PPS-invariant and ergodic random field on a homogeneous tree. YangandLiu[8] have studiedastronglaw of largenumbersforthefrequencyofoccurrence of states for Markov chains field on a homogeneous tree (a particular case of tree-indexed Markov chains field and PPS-invariant random fields). Yang (see [6]) has studied the

4 22 Wang Kangkang, Zong Decai strong law of large numbers for frequency of occurrence of state and Shannon-McMillan theorem for homogeneous Markov chains indexed by a homogeneous tree. Recently, Yang (see [3]) has studied the strong law of large numbers and Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Huang and Yang (see []) have also studied the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree. Peng and Yang have studied a class of small deviation theorems for functionals for arbitrary random field on a homogeneous trees (see[9]). Wang has also studied some Shannon-McMillan approximation theorems for arbitrary random field on the generalized Bethe tree (see[0]). In this paper, we study a class of Shannon-McMillan random approximation theorems for arbitrary random fields on the double rooted tree by comparison the arbitrary measure with the second-order nonhomogeneous Markov measure and constructing a supermartingale on the double rooted tree. As corollaries, a class of Shannon-McMillan theorems for arbitrary random field and second-order Markov chains field on the double rooted tree are obtained. A limit property for the expectation of the random conditional entropy of second-order homogeneous Markov chain indexed by the double rooted tree is studied. Yang and Ye s result (see[3]) is extended. 2. Main result and its proof Lemma (see [8]). Let µ and µ 2 be two probability measures defined on (Ω,F), D F, {τ n,n 0} be a sequence of positive-valued random variables such that n τ n > 0. µ a.s. D. (0) Then log µ 2(X T(n) ) τ n µ (X T(n) ) 0. µ a.s. D. () In particular, let τ n =, then log µ 2(X T(n) ) µ (X T(n) ) 0. µ a.s. (2) Proof. See reference [8]. Let ϕ(µ µ Q ) = limsup log µ(xt(n) ) µ Q (X T(n) ), (3)

5 ϕ(µ µ Q ) is called the sample relative entropy rate of X T(n) with respect to µ and µ Q. ϕ(µ µ Q ) is also called asymptotic logarithmic likelihood ratio. By (2) and (3) ϕ(µ µ Q ) liminf 23 µ(xt(n) ) log 0. µ a.s. (4) µ Q (X T(n) ) Hence ϕ(µ µ Q ) can be looked on as a type of a measure of the deviation between the arbitrary random field and the second-order nonhomogeneous Markov chain fields on the double rooted tree. Although ϕ(µ µ Q ) is not a proper metric between two probability measures, we nevertheless think of it as a measure of dissimilarity between their joint distribution µ and second-order Markov distribution µ Q. Obviously, ϕ(µ µ Q ) = 0 if and only if µ = µ Q. It has been shown in (4) that ϕ(µ µ Q ) 0, a.s. in any case. Hence, ϕ(µ µ Q ) can be used as a random measure of the deviation between the true joint distribution µ(x T(n) ) and the second-order Markov distribution µ Q (x T(n) ). Roughly speaking, this deviation may be regarded as the one between coordinate stochastic process x T(n) and the Markov case. The smaller ϕ(µ µ Q ) is, the smaller the deviation is. Theorem. Let X = {X t,t T} be an arbitrary random field on a double rooted tree. f n (ω) and ϕ(µ µ Q ) are respectively defined as (7) and (3). Denote by H Q t (X t X t,x 2t ) the random conditional entropy of X t relative to X t,x 2t on the measure µ Q, that is H Q t (X t X t,x 2t ) = x t SQ t (x t X t,x 2t )logq t (x t X t,x 2t ), t T (n) \{o, }. (5) Let Then D(c) = {ω : ϕ(µ µ Q ) c} (6) { 2xe 2 α(c) = min ( x) 2 + c } x, 0 < x <, c > 0; α(0) = 0. (7) { 2xe 2 β(c) = max (+x) 2 + c } x, < x < 0, c > 0; β(0) = 0. (8) [f n (ω) [f n(ω) H Q t (X t X t,x 2t )] α(c)m, µ a.s. ω D(c) (9) H Q t (X t X t,x 2t )] β(c)m c. µ a.s. ω D(c) (20)

6 24 Wang Kangkang, Zong Decai Proof. Consider the probability space (Ω,F,µ), let λ > 0 be a constant, δ j ( ) be Kronecker function. Denote g t (j) = logq t (j X t,x 2t ), we construct the following product distribution: µ Q (x T(n) ;λ) = q(x 0,x ) By (22) we can write = x Ln S Ln q(x 0,x ) Q t (x t x t,x 2t ) exp{λg t (j)δ j (x t )}[ +(e λgt(j) )Q t (j x t,x 2t ) ]. µ Q (x T(n) ;λ) x Ln S Ln Q t (x t x t,x 2t ) exp{λg t (j)δ j (x t )}[ +(e λgt(j) )Q t (j x t,x 2t ) ] Q = µ Q (x T(n ) t (x t x t,x 2t ) ;λ) exp{λg t (j)δ j (x t )}[ +(e λgt(j) )Q x Ln S Ln t L t (j x t,x 2t ) ] n = µ Q (x T(n ) ;λ) Q t (x t x t,x 2t ) exp{λg t (j)δ j (x t )}[ +(e λgt(j) )Q t (j x t,x 2t ) ] = µ Q (x T(n ) ;λ) t L n x t S = µ Q (x T(n ) ;λ) +(e λgt(j) )Q t L t (j x t,x 2t ) [ n x t=j + x t j eλgt(j) Q t (j x t,x 2t)+ Q t (j x t,x 2t) +(e λgt(j) )Q t L t (j x t,x 2t ) n ] (22) = µ Q (x T(n ) ;λ) (23) Therefore µ Q (x T(n) ;λ), n =,2, are a class of consistent distributions on S T(n). Let U n (λ,ω) = µ Q(X T(n) ;λ) µ(x T(n) ) (24) By (22) and (24) we attain U n (λ,ω) = exp{ q(x 0 ) λg t (j)δ j (X t )} Q t (X t X t,x 2t ) [ +(e λgt(j) )Q t (j X t,x 2t ) ] / µ(x T(n) ). (25) It is easy to see that U n (λ,ω) is a nonnegative sup-martingale from Doob s martingale convergence theorem (see [2]) since µ and µ Q are two probability measures. Moreover, lim U n(λ,ω) = U (λ,ω) <. µ a.s. (26)

7 25 By (2) and (24) we have According to (6), (25), we can rewrite (27) as { Letting λ = 0 in (28), we have ϕ(µ µ Q ) liminf logu n(λ,ω) 0. µ a.s. (27) λg t (j)δ j (X t ) log[+(e λgt(j) )Q t (j X t,x 2t )]+ log µ Q(X T(n) ) µ(x T(n) ) } 0 µ a.s. (28) By use of (6) and (28) we obtain µ(xt(n) ) log 0. µ a.s. ω D(c). (29) µ Q (X T(n) ) {λg t (j)δ j (X t ) log[+(e λgt(j) )Q t (j X t,x 2t )} ϕ(µ µ Q ) c. µ a.s. ω D(c). (30) By virtue of (30), the properties of super limit and the inequalities /x lnx x,(x > 0), e x x (/2)x 2 e x, we can write = (λ 2/ 2)limsup (λ 2/ 2)limsup λ{g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )} {log[+(e λgt(j) )Q t (j X t,x 2t )] λg t (j)q t (j X t,x 2t )}+c Q t (j X t,x 2t )[e λgt(j) λg t (j)]+c Q t (j X t,x 2t )g 2 t(j)e λgt(j) +c Q t (j X t,x 2t )log 2 Q t (j X t,x 2t )e λ logqt(j X t,x 2t ) +c

8 26 Wang Kangkang, Zong Decai = (λ 2/ 2)limsup log 2 Q t (j X t,x 2t ) Q t (j X t,x 2t ) λ +c. µ a.s. ω D(c) (3) In the case 0 < λ <, dividing two sides of (3) by λ, we have λ 2 limsup Consider the function [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] log 2 Q t (j X t,x 2t ) Q t (j X t,x 2t ) λ + c λ µ a.s. ω D(c) (32) φ(x) = (logx) 2 x λ, 0 < x, 0 < λ <. (set φ(0) = 0) (33) It can be concluded that on the internal [0,], max{φ(x),0 x } = φ(e 2/(λ ) 2 ) = ( λ )2 e 2. (34) By (32) and (34) we have that when 0 < λ <, λ 2 limsup = 2λe 2 ( λ) 2 limsup [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] 2 ( λ )2 e 2 + c λ 2 T (n) + c λ 2λe 2 ( λ) 2+c. µ a.s. ω D(c) (35) λ When c > 0, h(λ) = (2λe 2 ) / ( λ) 2 +c/λ attains its smallest value α(c) at λ o (0,). Hence letting λ = λ o in (35), we attain from (7) that [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] α(c). µ a.s. ω D(c). (36) By (7), (6), (5), (29) and (36), noticing g t (j) = logq t (j X t,x 2t ), we can deduce [f n (ω) H Q t (X t X t,x 2t )]

9 = limsup{ logµ(xt(n) T (n) ) +limsup j S [ t T (n) \{o} H Q t (X t X t )} { logq t (X t X t,x 2t ) H Q t (X t X t,x 2t )} logq t (X t X t,x 2t ) logµ(x T(n) )] = limsup { δ j (X t )logq t (j X t,x 2t )+Q t (j X t,x 2t )logq t (j X t,x 2t )} 27 = limsup le s M limsup j=s +limsup +limsup log µ Q(X T(n) ) µ(x T(n) ) s M j=s [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] log µ Q(X T(n) ) µ(x T(n) ) [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] α(c)m µ a.s. ω D(c), (37) thus in the case c > 0, (9) follows from (37). In the case < λ < 0, by (3) we have λ 2 limsup By (34) and (38), we gain λ 2 limsup 2λe 2 (+λ) 2+c λ. [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] log 2 Q t (j X t,x 2t ) Q t (j X t,x 2t ) +λ + c λ. µ a.s. ω D(c). (38) [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] 2 ( +λ )2 e 2 + c λ

10 28 Wang Kangkang, Zong Decai µ a.s. ω D(c). (39) In the case c > 0, the function u(λ) = (2λe 2 ) / (+λ) 2 +c/λ attains the largest value β(c) at λ o (,0). Thereby letting λ = λ o in (39), we have [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] β(c). µ a.s. ω D(c). (40) By (7), (6), (3), (6) and (40), noticing that g t (j) = logq t (j X t,x 2t ), we can write liminf s M [f n(ω) +liminf liminf liminf j=s [ limsup H Q t (X t X t,x 2t ) { logq t (X t X t,x 2t ) H Q t (X t X t,x 2t )} logq t (X t X t,x 2t ) logµ(x T(n) )] s M j=s [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] [logµ(xt(n) ) t T (n) \{o} logq t (X t X t )] [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] ϕ(µ µ Q ) β(c)m c. µ a.s. ω D(c). (4) In accordance with (4), we see that (20) also holds in the case c > 0. When c = 0, take 0 < λ i <,(i =,2, ) such that λ i 0 (i ), by (35) we acquire [g t (j)δ j (X t ) g t (j)q t (j X t,x 2t )] 0. µ a.s. ω D(0). (42) Imitating the proof of (37), we have by (7) and (42) [f n (ω) H Q t (X t X t,x 2t )] 0.µ a.s. ω D(0). (43)

11 Since α(0) = 0, we know that (9) also holds in the case c = 0 from (37). By the similar means, we can obtain that (20) holds in the case c = 0. Corollary. Under the assumption of Theorem, we have that in the case 0 c <, [f n (ω) T (n) H Q t (X 2e 2 t X t,x 2t )] M[ ( c) 2 +] c, [f n(ω) 29 µ a.s. ω D(c) (44) H Q t (X 2e 2 t X t,x 2t )] M[ ( c) 2 +] c c. Proof. Letting x = c in (9) and (20), we have µ a.s. ω D(c). (45) [2e 2/ ( c) 2 +] c α(c), [2e 2/ ( c) 2 +] c β(c). Therefore (44), (45) follow from (9) and (20), respectively. Corollary 2. Under the assumption of Theorem, we have lim [f n(ω) T (n) H Q t (X t X t,x 2t )] = 0. µ a.s. ω D(0). (46) Proof. Letting c = 0 in Corollary, (46) follows from (44) and (45). Corollary 3. Let X = {X t,t T} be the second-order nonhomogeneous Markov chains field indexed by the double rooted tree with the initial distribution (5) and the joint distribution (6), f n (ω) be defined as (8). Then lim [f n(ω) T (n) H Q t (X t X t,x 2t )] = 0. µ Q a.s. (47) Proof. Let µ µ Q in Theorem, then ϕ(µ µ Q ) 0. Thereby D(0) = Ω. (47) follows from (46) correspondingly. Remark. When the second-order nonhomogeneous Markov chain indexed by the tree degenerates into the first-order nonhomogeneous Markov chain indexed by a tree, we can see easily that Q t (X t X t,x 2t ) = Q t (X t X t ), H Q t (X t X t,x 2t ) = H Q t (X t X t ). At the moment, (47) is changed into lim [f n(ω) T (n) H Q t (X t X t )] = 0. µ Q a.s. This is a main result of Yang and Ye (see [3]).

12 30 Wang Kangkang, Zong Decai 3. Shannon-McMillan theorem for homogeneous Markov chains fields on a double rooted tree Let X = {X t,t T} be another second-order homogeneous Markov chain indexed by a double rooted tree with the initial distribution and the joint distribution on the measure µ P as follows: µ P (x 0,x ) = p(x 0,x ) (48) µ P (x T(n) ) = p(x 0,x ) P(x t x t,x 2t ), n, (49) where P = P(z x,y), x,y,z S is a strictly positive stochastic matrix on S 3, p = (p(x,y)) is a strictly positive distribution on S 2. Thereby the relative entropy density of X = {X t,t T} on the measure µ P is f n (ω) = [logp(x 0,X )+ logp(x t x t,x 2t )]. (50) Let a be a real number, denote [a] + = max{a,0}. We have the following result: Theorem 2. Let X = {X t,t T} be the second-order homogeneous Markov chains field with the initial distribution (48) and joint distribution (49) under the measure µ P. f n (ω) is defined by (50). Let Q = Q(z x,y), x,y,z S be defined by definition, α = min{q(j i,k), i,k,j S} > 0. Denote H Q (X t X t,x 2t ) = x t SQ(x t X t,x 2t )logq(x t X t,x 2t ), t T (n) \{o, }. If [P(j i,k) Q(j i,k)] + α c, (5) i S k S j S then [f n (ω) H Q (X t X t,x 2t )] α(c)m. µ P a.s. (52) [f n(ω) H Q (X t X t,x 2t )] β(c)m c. µ P a.s. (53) Proof. Let µ = µ P, Q t (z x,y) Q(z x,y), x,y,z S, t T (n) \{o, }, we obtain that H Q t (X t X t,x 2t ) = H Q (X t X t,x 2t ), t T (n) \{o, }, thus (50) follows from (7)

13 and (49). By the inequalities logx x (x > 0), a [a] + and (5), noticing that α = min{q(j i,k), i,k,j S} > 0, we can conclude = limsup ϕ(µ P µ Q ) = limsup log log µ P(X T(n) ) µ Q (X T(n) ) p(x 0,X ) P(X t X t,x 2t ) q(x 0,X ) log p(x 0,X ) q(x 0,X ) +limsup i S i S k S j S i S k S j S k S j S i S i S k S j S Q(X t X t,x 2t ) log P(X t X t,x 2t ) Q(X t X t,x 2t ) δ j (X t )δ i (X t )δ k (X 2t )log P(j i,k) Q(j i, k) P(j i,k) Q(j i,k) δ j (X t )δ i (X t )δ k (X 2t ) Q(j i, k) i S k S j S k S j S By (5) and (54) we have P(j i,k) Q(j i,k) Q(j i, k) [P(j i,k) Q(j i,k)] + α [P(j i,k) Q(j i,k)] + 2[P(j i,k) Q(j i,k)] + α = [P(j i,k) Q(j i,k)] +. (54) α i S k S j S ϕ(µ P µ Q ) c. a.s. (55) By (6) and (55) we know D(c) = Ω. Hence (52), (53) follow from (9), (20), respectively. Theorem 3. Let X = {X t,t T} be a second-order homogeneous Markov chains field indexed by the double rooted tree with the initial distribution () and the transition matrix Q = Q(z x,y), x,y,z S. Then lim E Q [H Q (X t X t,x 2t )] = i S α 3 q(i, j)q(k i, j) log Q(k i, j), j S k S

14 32 Wang Kangkang, Zong Decai µ Q a.s., (56) where E Q represents the expectation under the measure µ Q. Proof. By the definition of H Q (X t X t,x 2t ) in Theorem 2 and the property of the conditional expectation, we can write lim = lim = lim = lim = lim = lim E Q [H Q (X t X t,x 2t )] E Q [E Q ( logq(x t X t,x 2t ) X t,x 2t )] E Q ( logq(x t X t,x 2t )) x t S x 2t S x t S i S j S k S i S j S k S [ q(x t,x 2t,x t )logq(x t x t,x 2t )] [ q(i,j,k)logq(k i,j)] [ q(i, j)q(k i, j) log Q(k i, j)] = 2 [ q(i,j)q(k i,j)logq(k i,j)] lim i S j S k S = [ q(i, j)q(k i, j) log Q(k i, j)]. (57) i S j S k S Therefore, (56) follows from (57) directly. 4. Acknowledgements Authors would like to thank the referee for his valuable suggestions. The work is supported by the Project of Higher Schools Natural Science Basic Research of Jiangsu Province of China (3KJB0006). Wang Kangkang is the corresponding author. References [] I. Benjammini and Y. Peres, Markov chains indexed bu trees. Ann. Probab. 22(2): [2] T. Berger, and Z. Ye, Entropic aspects of random fields on trees. IEEE Trans. Inform. Theory. 36(3):

15 [3] R. Pemantle, Antomorphism invariant measure on trees. Ann. Probab. 20(3): [4] Z.X. Ye, and T. Berger, Ergodic regularity and asymptotic equipartition property of random fields on trees. J.Combin.Inform.System.Sci, 2(): [5] Z.X. Ye, and T. Berger, Information Measures for Discrete Random Fields. Science Press, New York [6] W.G. Yang Some limit properties for Markov chains indexed by homogeneous tree. Stat. Probab. Letts. 65(2): [7] W. Liu, and W.G. Yang, An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains. Stochastic Process. Appl, 6(), [8] W.G. Yang and W. Liu, Strong law of large numbers and Shannon-McMillan theorem for Markov chains fields on trees. IEEE Trans.Inform.Theory. 48(): [9] W.C. Peng, W.G. Yang and B. Wang, A class of small deviation theorems for functionals of random fields on a homogeneous tree. Journal of Mathematical Analysis and Applications. 36, [0] K.K. Wang and D.C. Zong Some Shannon-McMillan approximation theoems for Markov chain field on the generalized Bethe tree. Journal of Inequalities and Applications. Article ID 47090, 8 pages doi:0.55/20/ [] H.L. Huang and W.G. Yang Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree, Science in China, 5(2): [2] J.L. Doob Stochastic Process. Wiley, New York [3] W.G. Yang and Z.X. Ye The asymptotic equipartition property for nonhomogeneous Markov chains indexed by a homogeneous tree. IEEE Trans.Inform.Theory. 53(5): Wang Kangkang School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 22003, China wkk.cn@26.com Zong Decai Department of Computer Science and Engineering, Changshu Institute of Technology, Changshu 25500, China Received April 202 Accepted 7 October 203