Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 RESEARCH Open Access A note on the strong law of large numbers for Markov chains indexed by irregular trees Wei-cai Peng* * Correspondence: weicaipeng@126.com Department of Mathematics, Chaohu University, Chaohu, 238000, P.R China Abstract In this paper, a kind of an infinite irregular tree is introduced The strong law of large numbers and the Shannon-McMillan theorem for Markov chains indexed by an infinite irregular tree are established The outcomes generalize some known results on regular trees and uniformly bounded degree trees Keywords: Markov chain; tree; strong law of large numbers; AEP Introduction By a tree T we mean an infinite, locally finite, connected graph with a distinguished vertex o called the root and without loops or cycles We only consider trees without leaves That is, the degree (the number of neighboring vertices) of each vertex (except o) is required to be at least Let T be an infinite tree with root o, the set of all vertices with distance n from the root is called the nth generation (or nth level) of T We denote by T (n) the union of the first (n) the union from the mth to nth generations of T, by Ln the n generations of T, by T(m) subgraph of T containing the vertices in the nth generation For each vertex t, there is a unique path from o to t, and |t| for the number of edges on this path We denote the first predecessor of t by t , the second predecessor of t by t , and by nt the nth predecessor of t We also call t one of t ’s sons For any two vertices s and t, denote by s ≤ t, if s is on the unique path from the root o to t, denote by s ∧ t the vertex farthest from o satisfying s ∧ t ≤ s and s ∧ t ≤ t X A = {Xt , t ∈ A} and denote by |A| the number of vertices of A If each vertex on a tree T has m + neighboring vertices, we call it a Bethe tree TB,m ; if the root has m neighbors and the other vertices have m + neighbors on a tree T, we call it a Cayley tree TC,m Both the Bethe tree and the Cayley tree are called regular (or homogeneous) trees If the degrees of all vertices on a tree T are uniformly bounded, then we call T a uniformly bounded degree tree (see [] and []) Definition (see []) Let T be a locally finite, infinite tree, S be a finite state-space, {Xt , t ∈ T} be a collection of S-valued random variables defined on the probability space ( , F , P) Let p(x), x ∈ S () © 2014 Peng; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 Page of be a distribution on S, and P(y|x) , x, y ∈ S () be a stochastic matrix on S If for any vertex t, P(Xt = y|Xt = x and Xs for t ∧ s ≤ t ) = P(Xt = y|Xt = x) = P(y|x) ∀x, y ∈ S () and P(X = x) = p(x) ∀x ∈ S, then {Xt , t ∈ T} will be called S-valued Markov chains indexed by an infinite tree T with initial distribution () and transition matrix (), or called T-indexed Markov chains with state-space S Benjamini and Peres [] gave the notion of tree-indexed Markov chains and studied the recurrence and ray-recurrence for them Berger and Ye [] studied the existence of entropy rate for some stationary random fields on a homogeneous tree Ye and Berger [], by using Pemantle’s result [] and a combinatorial approach, studied the Shannon-McMillan theorem with convergence in probability for a PPG-invariant and ergodic random field on a homogeneous tree Yang and Liu [] studied the strong law of large numbers and Shannon-McMillan theorems for Markov chains field on the Cayley tree Yang [] studied some strong limit theorems for homogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the asymptotic equipartition property (AEP) for finite homogeneous Markov chains indexed by a homogeneous tree Yang and Ye [] studied strong theorems for countable nonhomogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the AEP for finite nonhomogeneous Markov chains indexed by a homogeneous tree Bao and Ye [] studied the strong law of large numbers and asymptotic equipartition property for nonsymmetric Markov chain fields on Cayley trees Takacs [] studied the strong law of large numbers for the univariate functions of finite Markov chains indexed by an infinite tree with uniformly bounded degree Huang and Yang [] studied the strong law of large numbers for Markov chains indexed by uniformly bounded degree trees However, the degrees of the vertices in the tree models are uniformly bounded What if the degrees of the vertices are not uniformly bounded? In this paper, we drop the uniformly bounded restriction We mainly study the strong law of large numbers and AEP with a.e convergence for finite Markov chains indexed by trees under the following assumption For any integer N ≥ , let d (t) := and denote dN (t) := |σ ∈ T : Nσ = t| by the amount of t’s N th descendants Denote O(n) = cn : < lim sup n→∞ cn ≤ c, c is a constant n () Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 Page of We assume that for enough large n and any given integer N ≥ , max dN (t) : t ∈ T (n) ≤ O ln |T (n+N) | |T (n) | () The following examples are used to explain assumption () Example Both the Bethe tree TB,m and the Cayley tree TC,m satisfy assumption () Actually, max{dN (t) : t ∈ T (n–N) } is a constant mN , and ln(|T (n) |/|T (n–N) |) = N ln m Example A uniformly bounded degree tree satisfies assumption () In fact, if the tree T is a uniformly bounded degree tree, then max{dN (t) : t ∈ T (n–N) } is no more than a constant aN , and ln |T (n) | |T (n–N) | × aN ≤ ln = N ln a |T (n–N) | |T (n–N) | is also a constant Example Define the lower growth rate of the tree to be gr T = lim infn→∞ |T (n) | n and the upper growth rate of the tree to be Gr T = lim supn→∞ |T (n) | n If both the gr T and Gr T are finite, then ln |T (n+N) | (Gr T)n+N Gr T ≤ ln + N ln Gr T, = n ln (n) n |T | (gr T) gr T hence () implies that max dN (t) : t ∈ T (n) ≤ O ln |T (n+N) | |T (n) | ≤ O(n) Some notations and lemmas In the following, let δk (·) be a Kronecker δ-function For any given integer N ≥ , denote δk (Xt )dN (t), SkN T (n) := () t∈T (n–N) which can be considered as the number of k’s among the variables in T (n–N) , weighted according to the number of N th descendants By (), we have SnN (k) = T (n–N) – () k∈S Define Hn (ω) = gt (Xt , Xt ) () E gt (Xt , Xt )|Xt () t∈T (n) \{o} and Gn (ω) = t∈T (n) \{o} Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 Page of Lemma (see []) Let T be an infinite tree with assumption () holds Let (Xt )t∈T be a T-indexed Markov chain with state-space S defined as before, {gt (x, y), t ∈ T} be functions (n) defined on S Let Lo = {o}, Fn = σ (X T ), λ g (X ,X ) t∈T (n) \{o} t t t e tn (λ, ω) = λgt (Xt ,Xt ) |Xt ] t∈T (n) \{o} E[e , () where λ is a real number Then {tn (λ, ω), Fn , n ≥ } is a nonnegative martingale Lemma (see []) Under the assumption of Lemma , let {an , n ≥ } be a sequence of nonnegative random variables, α > Set B= lim an = ∞ () n→∞ and D(α) = lim sup n→∞ an E gt (Xt , Xt )eα|gt (Xt ,Xt )| |Xt = M(ω) < ∞ ∩ B () t∈T (n) \{o} Then Hn (ω) – Gn (ω) = a.e on D(α) n→∞ an lim () Strong law of large numbers and Shannon-McMillan theorem In this section, we study the strong law of large numbers and the Shannon-McMillan theorem for finite Markov chains indexed by an infinite tree with assumption () holds Theorem Let T be an infinite tree with assumption () holds Then under the assumption of Lemma , for all k ∈ S and N ≥ , we have lim n→∞ |T (n+N) | SkN T (n) – SlN+ T (n–) P(k|l) = a.e () l∈S Proof Let gt (x, y) = dN (t)δk (y), an = |T (n+N) | Since Gn (ω) = E gt (Xt , Xt )|Xt t∈T (n) \{o} δk (xt )P(xt |Xt ) dN (t) = xt ∈S t∈T (n) \{o} dN (t)P(k|Xt ) = t∈T (n) \{o} δl (Xt )dN (t)P(k|l) = l∈S t∈T (n) \{o} δl (Xt )dN+ (t)P(k|l) = l∈S t∈T (n–) SlN+ T (n–) P(k|l) = l∈S () Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 Page of and Hn (ω) = dN (t)δk (Xt ) = SkN T (n) – δk (Xo )dN (o) gt (Xt , Xt ) = t∈T (n) \{o} () t∈T (n) \{o} By Lemma , we know that {tn (λ, ω), Fn , n ≥ } is a nonnegative martingale According to the Doob martingale convergence theorem, we have lim tn (λ, ω) = t(λ, ω) < ∞ a.e n () We have by () lim sup n→∞ ln tn (λ, ω) ≤ a.e ω ∈ B |T (n+N) | () By (), () and (), we get lim sup n→∞ λHn (ω) – |T (n+N) | ln E eλg(Xt ,Xt ) |Xt ≤ a.e ω ∈ B () t∈T (n) \{o} Let λ > Dividing two sides of () by λ, we have lim sup n→∞ Hn (ω) – an t∈T (n) \{o} ln[E[eλg(Xt ,Xt ) |Xt ]] ≤ a.e ω ∈ B λ () The case {dN (t) : t ∈ T (n) } is uniformly bounded was considered in [], we only consider the case {dN (t) : t ∈ T (n) } is not uniformly bounded By () and inequalities ln x ≤ x – (x > ), ≤ ex – – x ≤ – x e|x| , as < λ ≤ α, we have lim sup n→∞ Hn (ω) – |T (n+N) | ≤ lim sup n→∞ ≤ lim sup n→∞ ≤ = ≤ ≤ E gt (Xt , Xt )|Xt t∈T (n) \{o} ln[E[eλgt (Xt ,Xt ) |Xt ]] – E gt (Xt , Xt )|Xt λ |T (n+N) | t∈T (n) E[eλgt (Xt ,Xt ) |Xt ] – – E gt (Xt , Xt )|Xt λ |T (n+N) | t∈T (n) λ lim sup (n+N) n→∞ |T | λ lim sup n→∞ |T (n+N) | λ lim sup n→∞ |T (n+N) | λ lim sup n→∞ |T (n+N) | E gt (Xt , Xt )eλ|gt (Xt ,Xt )| |Xt t∈T (n) E dN (t)δk (Xt ) eλ|d N (t)δ (X )| k t (n) t∈T() N (t) N (t) dN (t) eλd (n) t∈T() dN (t) eλd (n) t∈T() P(k|Xt ) |Xt Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 ≤ λ lim sup n→∞ |T (n+N) | eλd N (t) Page of for enough large dN (t) (n) t∈T() ≤ λ |T (n) | – N (n) lim sup (n+N) max eλd (t) , t ∈ T() n→∞ |T | ≤ (n) λ |T (n) | – max{dN (t),t∈T() } lim sup (n+N) e n→∞ |T | λ () By (), there exists a constant β > such that (n) max dN (t), t ∈ T() ≤ β ln |T (n+N) | , |T (n) | hence, (n) max{dN (t),t∈T() } λ e Set < λ < , β lim sup n→∞ < |T (n+N) | |T (n) | λβ () by () and () we have |T (n+N) | Hn (ω) – Gn (ω) λ |T (n) | – ≤ lim sup × |T (n+N) | n→∞ |T (n+N) | |T (n) | λβ () Let λ → + in (), by () and () we have lim n→∞ |T (n+N) | SlN+ T (n–) P(k|l) ≤ a.e SkN T (n) – () l∈S Let – β ≤ λ → – By (), we similarly get lim n→∞ SN T (n) – |T (n+N) | k SlN+ T (n–) P(k|l) ≥ a.e () l∈S Combining () and (), we obtain () directly Let T be a tree, (Xt )t∈T be a stochastic process indexed by the tree T with state-space S Denote P xT (n) = P XT (n) = xT (n) Let fn (ω) = – (n) ln P X T , |T (n) | () (n) fn (ω) will be called the entropy density of X T If (Xt )t∈T is a T-indexed Markov chain with state-space S defined by Definition , we have by () fn (ω) = – ln P(X ) + |T (n) | ln P(Xt , Xt ) t∈T (n) \{o} () Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 Page of The convergence of fn (ω) to a constant in a sense (L convergence, convergence in probability, a.e convergence) is called the Shannon-McMillan theorem or the entropy theorem or the AEP in information theory Theorem Let T be an infinite tree with assumption () holds Let k ∈ S, and P be an ergodic stochastic matrix Denote the unique stationary distribution of P by π Let (Xt )t∈T be a T-indexed Markov chain with state-space S generated by P Then, for given integer N ≥ , SkN (T (n) ) = π(k) n→∞ |T (n+N) | lim a.e () Let Sl,k (T (n) ) := |{t ∈ T (n) : (Xt , Xt ) = (l, k)}|, then Sl,k (T (n) ) = π(l)P(k|l) a.e n→∞ |T (n) | () lim Let fn (ω) be defined as (), then π(l)P(k|l) ln P(k|l) a.e lim fn (ω) = – n→∞ () l∈S k∈S Proof The proofs of () and () are similar to those of Huang and Yang ([], Theorem and Corollary ) Letting gt (x, y) = – ln P(y|x) in Lemma , then Hn (ω) = – lim (n) n→∞ |T | n→∞ |T (n) | lim fn (ω) = lim n→∞ n→∞ |T (n) | δl (Xt )δk (Xt ) ln P(k|l) = – lim t∈T (n) l∈S k∈S ln P(k|l) = – lim n→∞ ln P(Xt , Xt ) t∈T (n) \{o} l∈S k∈S Sl,k (T (n) ) |T (n) | by (), () holds Competing interests The author declares that they have no competing interests Acknowledgements This work is supported by the Foundation of Anhui Educational Committee (No KJ2014A174) Received: February 2014 Accepted: June 2014 Published: 23 June 2014 References Takacs, C: Strong law of large numbers for branching Markov chains Markov Process Relat Fields 8, 107-116 (2001) 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 51(2), 195-202 (2008) Benjamini, I, Peres, Y: Markov chains indexed by trees Ann Probab 22, 219-243 (1994) Berger, T, Ye, Z: Entropic aspects of random fields on trees IEEE Trans Inf Theory 36, 1006-1018 (1990) Ye, Z, Berger, T: Ergodic, regular and asymptotic equipartition property of random fields on trees J Comb Inf Syst Sci 21, 157-184 (1996) Pemantle, R: Automorphism invariant measure on trees Ann Probab 20, 1549-1566 (1992) Peng Journal of Inequalities and Applications 2014, 2014:244 http://www.journalofinequalitiesandapplications.com/content/2014/1/244 Yang, WG, Liu, W: Strong law of large numbers and Shannon-McMillan theorem for Markov chains field on Cayley tree Acta Math Sci Ser B 21(4), 495-502 (2001) Yang, WG: Some limit properties for Markov chains indexed by a homogeneous tree Stat Probab Lett 65, 241-250 (2003) Yang, WG, Ye, Z: The asymptotic equipartition property for nonhomogeneous Markov chains indexed by a homogeneous tree IEEE Trans Inf Theory 53(9), 3275-3280 (2007) 10 Bao, ZH, Ye, Z: Strong law of large numbers and asymptotic equipartition property for nonsymmetric Markov chain fields on Cayley trees Acta Math Sci Ser B 27(4), 829-837 (2007) doi:10.1186/1029-242X-2014-244 Cite this article as: Peng: A note on the strong law of large numbers for Markov chains indexed by irregular trees Journal of Inequalities and Applications 2014 2014:244 Page of ... Cite this article as: Peng: A note on the strong law of large numbers for Markov chains indexed by irregular trees Journal of Inequalities and Applications 2014 2014:244 Page of ... numbers and Shannon-McMillan theorem In this section, we study the strong law of large numbers and the Shannon-McMillan theorem for finite Markov chains indexed by an infinite tree with assumption ()... theorems for homogeneous Markov chains indexed by a homogeneous tree and the strong law of large numbers and the asymptotic equipartition property (AEP) for finite homogeneous Markov chains indexed by