RESEARC H Open Access A class of small deviation theorems for the random fields on an m rooted Cayley tree Zhiyan Shi 1* , Weiguo Yang 1 , Lixin Tian 1 and Weicai Peng 2 * Correspondence: shizhiyan1984@126.com 1 Faculty of Science, Jiangsu University, Zhenjiang 212013, China Full list of author information is available at the end of the article Abstract In this paper, we are to establish a class of strong deviation theorems for the random fields relative to mth-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 mth-order nonhomogeneous Markov chains indexed by that tree. 2000 Mathematics Subject Classification: 60F15; 60J10. Keywords: strong deviation theorem, m rooted Cayley tree, mth-order nonho moge- neous Markov chain, Shannon-McMillan theorem 1. Introduction AtreeisagraphG ={T, E} which is connected and contains no circuits. Given any two vertices s, t(s ≠ t Î T), let σ t be the unique path connecting s and t.Definethe graph distance d (s, t) to be the number of edges contained in the path σ t . Let T C,N be a Cayley tree. In this tree, the root (denoted by o)hasonlyN neighbors and all other vertices have N + 1 neighbors. Let T B, N be a Bethe tree, on which each ver- tex 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 T C,N (see Figure 1). It is formed by a Cayley tree T C,N with the root o co nnecting with anoth er 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 con- nected. When the context permits, this type of tree is denoted simply by T. Let s, t(s, t ≠ o, -1, - 2, , - (m - 1)) be vertices of an m rooted Cayler tree T. Write t ≤ s if t is on the unique path connecting o to s, and |s | the number of edges on this path. For any two vertices s, t(s, t ≠ o, -1, - 2, , - (m -1))oftreeT,denotebys ∧ t the vertex farthest from o satisfying s ∧ t ≤ s and s ∧ 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 espe- cially root -1 is on the -1st level on tree T, root -2 is on the -2nd level. By analogy, root -(m -1)isonthe-(m - 1) th le vel. We denote by T (n) the subtree of an m rooted Cayley tree T containing the vertices from level -(m -1)(theroot-(m -1))toleveln. 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)tot . We denote the predecessor of t by 1 t , the predecessor of 1 t by 2 t and the Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 © 2012 Shi et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/l icenses/by/2.0), which permits unre stricted use, distribution, and reproduction in any medium, provided the original work is properly ci ted. predecessor of (n -1) t by n t . We also say that n t is the n-th predecessor of t. X A ={X t , t Î A} is a stochastic process indexed by a set A, and denoted by |A|thenumberof vertices of A, x A is the realization of X A . Let (, F ) be a measure space, {X t , tÎT} be a collection of random variables defined on (, F ) and taking values in G = {0,1, , b - 1} , wh ere b isapositiveinteger.LetP be a general probability distribution on (, F ) . We will call P the random field on tree T. Denote the distribution of {X t , t Î T} under the probability measure P by P( x T (n) )=P(X T (n) = x T (n) ), x T (n) ∈ G T (n) . (1) Let f n (ω)=− 1 |T (n) | ln P(X T (n) ). (2) f n (ω) is called entropy density of X T (n) . Let Q be another probability measure on the measurable space (, F ) ,andletthe distribution of {X t , t Î T} under Q be Q(x T (n) )=Q(X T (n) = x T (n) ), x T (n) ∈ G T (n) . (3) Let h(P |Q) = lim sup n→∞ 1 |T (n) | ln P( X T (n) ) Q(X T (n) ) . (4) 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 appro ach 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}andP(y|x 1 , x 2 , , x m ) be a nonnegative functions on G m+1 . Let ❅ ❅ ❅ ❅ ❅      . . . level 0 root o level −1root−1 level −(m − 2) root −(m − 2) level − ( m − 1 ) root − ( m − 1 ) level −2root−2 level 2 level 3 ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ level 1 2 t ❆ ❆ ✁ ✁ ❆ ❆ ✁ ✁ ✁ ✁ ❆ ❆ ✁ ✁ ❆ ❆ 1 t ❇ ❇ ✂ ✂ ❇ ❇ ✂ ✂ ✂ ✂ ❇ ❇ ✂ ✂ ❇ ❇ ❇ ❇ ✂ ✂ ❇ ❇ ✂ ✂ ✂ ✂ ❇ ❇ ✂ ✂ ❇ ❇ t Figure 1 An m rooted Cayley tree ¯ T C,2 . Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 2 of 15 ¯ T C,2 If  y∈G P( y|x 1 , x 2 , , x m )=1, 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 Î T} be a collection of G-valued random variables defined on the probability space (, F , Q) .LetQ be a probability on a measurable space (, F ) . Let q =(q(x 1 , x 2 , , x m )), x 1 , x 2 , , x m ∈ G (5) be a distribution on G m , and Q n =(q n (y|x 1 , x 2 , , x m )), x 1 , x 2 , , x m , y ∈ G, n ≥ 1 (6) be m-order transition matrices. For any vertex t Î L n , n ≥ 1, if Q(X t = y|X 1 t = x 1 , X 2 t = x 2 , , X m t = x m and X σ for σ ∧ t ≤ 1 t ) = Q( X t = y |X 1 t = x 1 , X 2 t = x 2 , , X m t = x m ) = q n (y|x 1 , x 2 , , x m ), ∀x 1 , x 2 , , x m , y ∈ G (7) and Q(X −(m−1) = x 1 , , X −1 = x m−1 , X o = x m ) = q ( x 1 , , x m−1 , x m ), x 1 , , x m ∈ G, (8) then {X t , t Î T} is called a G-valued mth-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 mth-order nonhomogeneous Markov chain under the probability measure Q. We denote o m = {o, −1, −2, , −(m −1)}, o  m = {−1, −2, , −(m − 1)}, X n 1 (t )={X n t , , X 2 t , X 1 t }, X n 0 (t )={X n t , ···, X 2 t , X 1 t , X t }, and denote by x n 1 (t ) and x n 0 (t ) the realizations X n 1 (t ) and X n 0 (t ) , respectively. Let {X t , t Î T}beanmth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tre e T under the probability measure Q defined on above. It is easy to see that Q(x T (n) )=Q(X T (n) = x T (n) )=q(x −(m−1) , , x o ) n  k=1  t∈L k q k (x t |x m 1 (t )). (9) In the following, we always assume that P(x T ( n )), Q(x T ( n )), q(x 1 , , x m ), and {q n (y | x 1 , , x m ), n ≥ 1} are all positive. Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 3 of 15 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 Marko v chains and studied the recurren ce and ray-recurrence for them. Berger and Ye [4] have studied the existence of en tropy rate for some stationary random fields on a homogeneous tree. Pemantle [5] proved a mixing prop erty and a weak law of large numbers for a PPG-invariant and ergodic random field on a homogeneou s tree. Ye and Berger [6,7], by using Pemantle’ s result and a combin atorial 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 infi- nite tree with uniformly bounded degree. Recently, Shi and Yang [12] have also studied some limit properties of random transition probability for second-order nonhomoge- neous 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 index ed by a homogeneous tree. Shi and Yang [2] have studied the strong law of large numbers and Shannon-McMillan for the mth-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. Yang [14] has also studi ed a class of small devia- tion theorems for the sequences of N-valued random variables with respect to mth- order nonhomogeneous Markov chains. In this paper, our main purpose is to extend Yang’s[14]resulttoanm rooted Cayley tree. By introducing the sample divergence rate of any probability measure with respect to mth-order nonhomogeneous Mar kov measure on an m rooted Cayley tree, we estab- lish a class of strong deviation theorems for the arbitrary random fields in dexed by that tree with respect to mth-order nonhomogeneous Markov chains indexed by that tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMilla n theo- rem for mth-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 roote d Cayley tree, G = {0, 1, , b -1}bethefinitestate space. Let {X t , t Î T} be a collection of G-valued random variables defined on the mea- surable space (, F ) .LetP and Q be two probability measures on the measurable space (, F ) ,andlet{X t , t Î T}beanmth-order nonhomogeneous Markov chains indexed by tree T under probability measure Q.Let{g n (y 1 , , y m+1 ), n ≥ 1} be a sequence of functions defined on G m+1 . Let F n = σ (X T (n) )(n ≥ 1) . Set F n (ω)= n  k=1  t∈L k g k (X m 0 (t )) (10) Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 4 of 15 and t n (λ, ω)= e λF n (ω)  n k=1  t∈L k E Q  e λg k (X m 0 (t)) |X m 1 (t)  · q(X −(m−1) , , X o )  n k=1  t∈L k q k (X t |X m 1 (t)) P(x T (n) ) , (11) where E Q denote the expectation under probability measure Q.Then {t n (λ, ω), F n , n ≥ 1} 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 Î T} be a collection of random var iables taking values in G = {0, 1, , b -1}definedonthemeasurablespace (, F ) . Let P an d Q be two probability measures on the measurable space (, F ) ,suchthat {X t , t Î T}isanmth-order nonhomogeneous Markov chain indexed by T under Q. Let h(P | Q) be defined by (4), {g n (y 1 , , y m+1 ), n ≥ 1} be a sequence of functions defined on G m+1 . Let c ≥ 0 be a constant. Set D(c)={ω : h(P|Q) ≤ c}. (12) Assume that there exists a >0, such that ∀i m Î G m , b α (i m ) = lim sup n→∞ 1 |T (n) | n  k=1  t∈L k E Q [e a|g k (X m 0 (t))| |X m 1 (t )=i m ] ≤ τ . (13) Let A t = 2τ e 2 (t − α) 2 , (14) where o<t<a. Thus, when 0 ≤ c ≤ t 2 A t , we have lim sup n→∞ 1 |T (n) |       n  k=1  t∈L k {g k (X m 0 (t)) −E Q [g k (X m 0 (t))|X m 1 (t)]}       lim sup ≤ 2  cA t , P−a.e., ω ∈ D(c). (15) In particular, lim n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t)) − E Q [g k (X m 0 (t))|X m 1 (t)]} =0, P − a.e., ω ∈ D(0). (16) Proof Let t n ( l, ω) be defined by (11). By Lemma 1, {t n (λ, ω), F n , n ≥ 1} is a non- negative martingale under probability measure P.ByDoob’s martingale convergence theorem, we have lim n→∞ t n (λ, ω)=t(λ, ω) < ∞, P − a.e. Hence, lim sup n→∞ 1 |T (n) | ln t n (λ, ω) ≤ 0, P − a.e (17) We have by (9), (10), (11) and (17) lim sup n→∞ 1 |T (n) | ⎡ ⎣ n  k=1  t∈L k  λg k (X m 0 (t)) − ln E Q  e λg k (X m 0 (t)) |X m 1 (t)  − ln P(X T (n) ) Q(X T(n) ) ⎤ ⎦ ≤ 0, P−a.e. (18) Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 5 of 15 By (4),(12) and (18) lim sup n→∞ 1 |T (n) | n  k=1  t∈L k  λg k (X m 0 (t)) − ln E Q  e λg k (X m 0 (t)) |X m 1 (t)  ≤ c, P−a.e., ω ∈ D(c). (19) This implies that lim sup n→∞ λ |T (n) | n  k=1  t∈L k {g k (X m 0 (t)) −E Q [g k (X m 0 (t))|X m 1 (t)]} ≤ lim sup n→∞ 1 |T (n) | n  k=1  t∈L k  ln E Q  e λg k (X m 0 (t)) |X m 1 (t)  − E Q [λg k (X m 0 (t))|X m 1 (t)]  + c, P − a.e., ω ∈ D(c) (20) Let |l| <t.ByinequalitiesInx ≤ x -1(x>0) and e x − 1 −x ≤ x 2 2 e |x| , and noticing that max{x 2 e −hx , x ≥ 0} =4e −2 /h 2 (h > 0). (21) We have lim sup n→∞ 1 |T (n) | n  k=1  t∈L k  ln E Q  e λg k (X m 0 (t)) |X m 1 (t)  − E Q [λg k (X m 0 (t))|X m 1 (t)]  ≤ lim sup n→∞ 1 |T (n) | n  k=1  t∈L k  E Q  e λg k (X m 0 (t)) |X m 1 (t)  − 1 − E Q [λg k (X m 0 (t))|X m 1 (t)]  ≤ λ 2 2 lim sup n→∞ 1 |T (n) | n  k=1  t∈L k E Q  g k 2 (X m 0 (t))e |λ|| g k (X m 0 (t))| |X m 1 (t)  = λ 2 2 lim sup n→∞ 1 |T (n) | n  k=1  t∈L k E Q  e α|g k (X m 0 (t))| g k 2 (X m 0 (t))e (|λ|−α)|g k (X m 0 (t))| |X m 1 (t)  ≤ λ 2 2 lim sup n→∞ 1 |T (n) | n  k=1  t∈L k E Q  e α|g k (X m 0 (t))| 4e −2 /(|λ|−a) 2 |X m 1 (t)  ≤ 2λ 2 τ /e 2 (t −α) 2 . (22) By (20) and (22), we have lim sup n→∞ λ |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≤ λ 2 A t + c, P − a .e ., ω ∈ D(c). (23) When 0 < l <t<a, we have by (23) lim sup n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≤ λA t + c/λ, P − a.e., ω ∈ D(c). (24) It is easy to see that when 0 <c<t 2 A t ,thefunctionf (l)=lA t + c/ l attains, at λ =  c/A t , its smallest value f (  c/A t )=2 √ cA t . Letting λ =  c/A t in (24), we have lim sup n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t)) − E Q [g k (X m 0 (t))|X m 1 (t)]}≤2  cA t , P−a.e., ω ∈ D(c). (25) Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 6 of 15 When c = 0, we have by (24) lim sup n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t)) − E Q [g k (X m 0 (t))|X m 1 (t)]}≤λA t , P−a.e., ω ∈ D(0). (26) Letting l ® 0 + in (26), we obtain lim sup n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t)) − E Q [g k (X m 0 (t))|X m 1 (t)]}≤0, P −a.e., ω ∈ D(0). (27) Hence, (25) also holds for c =0.When-a <-t<l <0, by virtue of (23) it can be shown in a similar way that lim inf n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t)) − E Q [g k (X m 0 (t))|X m 1 (t)]}≥−2  cA t , P−a.e., ω ∈ D(c). (28) Equation 15 follows from (25) and (28), Equation 15 implies (16) immediately. This completes the proof of the theorem. □ Theorem 2 Let H t =2b/e 2 (t − 1) 2 ,0< t < 1. (29) Let f n (ω ) be defined by (2). Under the conditions of Theorem 1, when 0 ≤ c ≤ t 2 H t , we have lim sup n→∞ {f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t )), , q k (b − 1|X m 1 (t ))]} ≤ 2  cH t , P − a.e., ω ∈ D(c), (30) lim inf n→∞ {f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t )), , q k (b − 1|X m 1 (t ))]} ≥−2  cH t − c, P − a.e ., ω ∈ D(c), (31) where H(p 0 , p b-1 ) denote the entropy of distribution (p 0 , , p b-1 ), i.e., H(p 0 , , p b−1 )=− b−1  i=0 p i ln p i . Proof In Theorem 1, let g k (y 1 , , y m+1 )=-Inq k (y m+1 | y 1 , , y m ) and a = 1, we have E Q  e g k (X m 0 (t)) |X m 1 (t )=i m  =  j∈G e |−ln q k (j|i m )| q k (j|i m ) =  j∈G q k (j|i m )/q k (j|i m ) = b. (32) Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 7 of 15 Hence, ∀i m Î G m , b 1 (i m ) = lim sup n→∞ 1 |T (n) | n  k=1  t∈L k E Q  e g k (X m 0 (t)) |X m 1 (t )=i m  ≤ b. (33) Noticing that E Q [−ln q k (X t |X m 1 (t ))|X m 1 (t )] = −  j∈G q k (j|X m 1 (t )) ln q k (j|X m 1 (t )) = H[q k (0|X m 1 (t )), , q k (b − 1|X m 1 (t ))] . (34) When 0 ≤ c ≤ t 2 H t , we have by (34),(29) and (15) lim sup n→∞ ⎧ ⎨ ⎩ 1 |T (n) | n  k=1  t∈L k (−ln q k (X t |X m 1 (t))) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b − 1|X m 1 (t))] ⎫ ⎬ ⎭ ≤ 2  cH t , P −a.e., ω ∈ D(c). (35) lim inf n→∞ ⎧ ⎨ ⎩ 1 |T (n) | n  k=1  t∈L k (−ln q k (X t |X m 1 (t))) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b − 1|X m 1 (t))] ⎫ ⎬ ⎭ ≥−2  cH t , P −a.e., ω ∈ D(c). (36) By (35), (9) and h(P|Q) ≥ 0, lim sup n→∞ ⎧ ⎨ ⎩ f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b − 1|X m 1 (t))] ≤ lim sup n→∞ {− 1 |T (n) | ln P(X T (n) ) − 1 |T (n) | n  k=1  t∈L k (−ln q k (X t |X m 1 (t)) ⎫ ⎬ ⎭ + lim sup n→∞ ⎧ ⎨ ⎩ 1 |T (n) | n  k=1  t∈L k (−ln q k (X t |X m 1 (t)) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b − 1|X m 1 (t))] ⎫ ⎬ ⎭ ≤ 2  cH t , P − a.e., ω ∈ D(c). (37) By (36), (9) and (12), we have lim inf n→∞ ⎧ ⎨ ⎩ f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b −1|X m 1 (t))] ≥ lim inf n→∞ ⎧ ⎨ ⎩ − 1 |T (n) | ln P(X T (n) ) − 1 |T (n) | n  k=1  t∈L k (−ln q k (X t |X m 1 (t)) ⎫ ⎬ ⎭ + lim inf n→∞ ⎧ ⎨ ⎩ 1 |T (n) | n  k=1  t∈L k (−ln q k (X t |X m 1 (t)) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b −1|X m 1 (t))] ⎫ ⎬ ⎭ ≥−h(P|Q) −2  cH t ≥−2  cH t − c, P − a.e., ω ∈ D(c). (38) Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 8 of 15 This completes the proof of this theorem. □ Corollary 1 Under the conditions of Theorem 2, we have lim n→∞ ⎧ ⎨ ⎩ f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b −1|X m 1 (t))] ⎫ ⎬ ⎭ =0, P−a.e., ω ∈ D(0). (39) If P<<Q, then lim n→∞ ⎧ ⎨ ⎩ f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b − 1|X m 1 (t))] ⎫ ⎬ ⎭ =0, P − a.e. (40) In particular, if P = Q, lim n→∞ ⎧ ⎨ ⎩ f n (ω) − 1 |T (n) | n  k=1  t∈L k H[q k (0|X m 1 (t)), , q k (b − 1|X m 1 (t))] ⎫ ⎬ ⎭ =0, Q − a.e. (41) 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 (y 1 , y m+1 ), n ≥ 1} is uniformly bounded, i.e., there exists M>0suchthat|g n (y 1 , , y m+1 )| ≤ M, then when c ≥ 0, we have lim sup n→∞ 1 |T (n) | | n  k=1  t∈L k {g k (X m 0 (t)) − E Q [g k (X m 0 (t))|X m 1 (t)]}| ≤ M(c+2 √ c), P−a.e., ω ∈ D(c). (42) Proof By (20) and (12) and the formula in line 2 of (22), we have lim sup n→∞ λ |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≤ lim sup n→∞ 1 |T (n) | n  k=1  t∈L k E Q [e λg k (X m 0 (t)) − 1 −λg k (X m 0 (t ))|X m 1 (t )] +cP− a.e., ω ∈ D(c). (43) By the hypothesis of the theorem and the inequality e x -1-x ≤ |x|(e |x| - 1), we have e λg k (X m 0 (t)) − 1 −λg k (X m 0 (t )) ≤|λ|M(e |λ|M − 1). (44) By (43) and (44) lim sup n→∞ λ |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≤|λ|M(e |λ||M − 1) + c, P − a.e., ω ∈ D(c). (45) When l >0, we have by (45) lim sup n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≤ M( e λM − 1) + c/λ, P − a.e., ω ∈ D(c). (46) Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 9 of 15 Taking λ = 1 M log(1 + √ c) , and using the inequality log(1 + √ c) ≥ √ c 1+ √ c , (47) we have when c>0 lim sup n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≤ M √ c + cM log(1 + √ c) ≤ M(2 √ c + c), P −a.e., ω ∈ D(c). (48) When l <0, it follows from (45) that lim inf n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≥−M(e λM − 1) + c/λ P − a.e., ω ∈ D(c). (49) Taking λ = − 1 M log(1 + √ c) in (49), and using (47), we have when c>0 lim inf n→∞ 1 |T (n) | n  k=1  t∈L k {g k (X m 0 (t )) − E Q [g k (X m 0 (t ))|X m 1 (t )]} ≥−M √ c − cM log(1 + √ c) ≥−M(2 √ c + c), P − a.e., ω ∈ D(c). (50) In a s imilar 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(y 1 , , y m+1 )beanyfunction defined on G m+1 . Let M = max g(y 1 , , y m+1 ). Then when c ≥ 0, lim sup n→∞ 1 |T (n) |       n  k=1  t∈L k {g(X m 0 (t)) − E Q [g(X m 0 (t))|X m 1 (t)]}       ≤ M(c+2 √ c), P−a.e., ω ∈ D(c). (51) Proof Letting g(y 1 , , y m+1 )=g n (y 1 , , y m+1 ), n ≥ 1 in Theorem 3, this corollary follows. In the following, let I k (x)=  1 x = k 0 x = k .Let S T (n) \o  m (i 1 , , i m ) be the number of (i 1 , , i m ) in the collection of {X m−1 0 (t ), t ∈ T (n) \o  m } , that is S T (n) \o  m (i 1 , , i m )= n  k=0  t∈L k I i 1 (X (m−1) t ) ···I i m (X t ), (52) S T (n) \o m (i 1 , , i m , i m+1 ) be the number of (i 1 , , i m , i m+1 ) in the collection of {X m 0 (t ), t ∈ T (n) \o m } , that is Shi et al. Journal of Inequalities and Applications 2012, 2012:1 http://www.journalofinequalitiesandapplications.com/content/2012/1/1 Page 10 of 15 [...]... nonhomogeneous Markov chains indexed by a tree J Inequal Appl ID 503203 (2009) 13 Peng, WC, Yang, WG, Wang, B: A class of small deviation theorems for functional of random fields on a homogeneous tree J Math Anal Appl 361, 293–301 (2009) 14 Yang, WG: A class of small deviation theorems for the sequences of N-valued random variables with respect to mthorder nonhomogeneous Markov chains Acta Mathematica Scientia... 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 Competing interests The authors declare that they have no competing interests Received: 21 March 2011 Accepted: 4 January 2012 Published: 4 January 2012 References 1 Liu, W, Wang, LY: The Markov approximation of the random field on Cayley tree and a class. .. (q(j|im )), j ∈ G, im ∈ Gm (61) be an mth-order transition matrix Define a stochastic matrix as follows: ¯ Q1 = (q(jm |im )), i m , j m ∈ Gm , (62) where q(jm |im ) = q(jm |im ), if jv = iv+1 , v = 1, 2, , m − 1, 0, otherwise (63) ¯ Then Q1 is called an m- dimensional stochastic matrix determined by the mth-order transition matrix.Q1 ¯ Lemma 2 (see [16]) Let Q1 be an m- dimensional stochastic matrix... 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) Author details 1 Faculty of Science, Jiangsu University, Zhenjiang 212013, China 2Department of Mathematics, Chaohu University, Chaohu 238000, China Authors’ contributions ZS, WY and LT carried... t∈Lk j∈G im ∈Gm = −N ST (n−1) \o m (im )q(j|im ) ln q(j|im ) j∈G im ∈Gm Noticing that limn→∞ |T (n) | | = |T (n−1) 1 N, by (39), (73) and (66), Equation 68 follows.□ 3 Shannon-McMillan Theorem Theorem 5 Let {Xt, t Î T} be a G-valued mth-order nonhomogeneous Markov chain indexed by an m rooted Cayley tree under the probability measure Q with initial distribution (5) and mth-order transition matrices (6)... 2 9A( 2), 517–527 (2009) in Chinese 15 Gray, RM: Entropy and Information Theory Springer, New York (1990) 16 Yang, WG, Liu, W: The asymptotic equipartition property for Mth-order nonhomogeneous Markov information sources IEEE Trans Inf Theory 50(12), 3326–3330 (2004) doi:10.1109/TIT.2004.838339 doi:10.1186/1029-242X-2012-1 Cite this article as: Shi et al.: A class of small deviation theorems for the random. .. (km )q(im |km ) = km ∈Gm By (59) and (69), we have lim n→∞ ST (n) \o m (im ) 1 − (n−1) |T (n) | |T | ST(n−1)\o m (km )q(im |km ) = 0, P a. e., ω ∈ D(0) (70) km ∈Gm Multiplying (70) by q(jm|im), adding them together for im Î Gm, and using (70) once again, we have q(jm |im ) · lim 0= n→∞ im ∈Gm = lim n→∞ + lim n→∞ = lim n→∞ im ∈Gm ST (n) \o m (im ) 1 − (n−1) |T (n) | |T | ST (n−1) \o m (km )q(im |km )... Entropic aspects of random fields on trees IEEE Trans Inf Theory 36, 1006–1018 (1990) doi:10.1109/ 18.57200 5 Pemantle, R: Andomorphism invariant measure on tree Ann Probab 20, 1549–1566 (1992) doi:10.1214/aop/1176989706 6 Ye, Z, Berger, T: Ergodic,regularity and asymptotic equipartition property of random fields on trees Combin Inf Syst Sci 21, 157–184 (1996) 7 Ye, Z, Berger, T: Information measure for. .. (n) \om (im ), ST (n) \om (im+1 ) and fn (ω) be defined as before Let Qn = Q1 = (q(j|im )), q(j|im ) > 0, ∀im ∈ Gm , j ∈ G, (74) ¯ be another positive mth-order transition matrix Let Q1 be an m dimension transi- tion matrix determined by Q1 If lim qn (j|im ) = q(j|im ), n→∞ ∀im ∈ Gm , j ∈ G, (75) then ST (n) \o m (im ) = π (im ), n→∞ |T (n) | lim P − a. e ST (n) \om (im+1 ) = π (im )q(im+1 |im ), n→∞... property for Markov chains indexed by a Homogeneous tree IEEE Trans Inf Theory 53(9), 3275–3280 (2007) 11 Huang, HL, Yang, WG: Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree Sci China Ser A Math 51(2), 195–202 (2008) doi:10.1007/s11425-008-0015-1 12 Shi, ZY, Yang, WG: Some limit properties of random transition probability for second-order nonhomogeneous . Wang, B: A class of small deviation theorems for functional of random fields on a homogeneous tree. J Math Anal Appl. 361, 293–301 (2009) 14. Yang, WG: A class of small deviation theorems for the. 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. this article as: Shi et al.: A class of small deviation theorems for the random fields on an m rooted Cayley tree. Journal of Inequalities and Applications 2012 2012:1. Shi et al. Journal of Inequalities