- Open Access
A note on the strong law of large numbers for Markov chains indexed by irregular trees
Journal of Inequalities and Applications volume 2014, Article number: 244 (2014)
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.
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 2.
Let T be an infinite tree with root o, the set of all vertices with distance n from the root is called the n th generation (or n th level) of T. We denote by the union of the first n generations of T, by the union from the m th to n th generations of T, by the subgraph of T containing the vertices in the n th generation. For each vertex t, there is a unique path from o to t, and for the number of edges on this path. We denote the first predecessor of t by , the second predecessor of t by , and by the n th predecessor of t. We also call t one of ’s sons. For any two vertices s and t, denote by , if s is on the unique path from the root o to t, denote by the vertex farthest from o satisfying and . and denote by the number of vertices of A.
If each vertex on a tree T has neighboring vertices, we call it a Bethe tree ; if the root has m neighbors and the other vertices have neighbors on a tree T, we call it a Cayley tree . 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 1 (see )
Let T be a locally finite, infinite tree, S be a finite state-space, be a collection of S-valued random variables defined on the probability space . Let
be a distribution on S, and
be a stochastic matrix on . If for any vertex t,
then will be called S-valued Markov chains indexed by an infinite tree T with initial distribution (1) and transition matrix (2), 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 , let and denote
by the amount of t’s N th descendants. Denote
We assume that for enough large n and any given integer ,
The following examples are used to explain assumption (5).
Example 1 Both the Bethe tree and the Cayley tree satisfy assumption (5). Actually, is a constant , and .
Example 2 A uniformly bounded degree tree satisfies assumption (5). In fact, if the tree T is a uniformly bounded degree tree, then is no more than a constant , and
is also a constant.
Example 3 Define the lower growth rate of the tree to be and the upper growth rate of the tree to be .
If both the grT and GrT are finite, then
hence (5) implies that
2 Some notations and lemmas
In the following, let be a Kronecker δ-function. For any given integer , denote
which can be considered as the number of k’s among the variables in , weighted according to the number of N th descendants. By (6), we have
Lemma 1 (see )
Let T be an infinite tree with assumption (5) holds. Let be a T-indexed Markov chain with state-space S defined as before, be functions defined on . Let , ,
where λ is a real number. Then is a nonnegative martingale.
Lemma 2 (see )
Under the assumption of Lemma 1, let be a sequence of nonnegative random variables, . Set
3 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 (5) holds.
Theorem 1 Let T be an infinite tree with assumption (5) holds. Then under the assumption of Lemma 1, for all and , we have
Proof Let , . Since
By Lemma 1, we know that is a nonnegative martingale. According to the Doob martingale convergence theorem, we have
We have by (17)
By (10), (16) and (18), we get
Let . Dividing two sides of (19) by λ, we have
The case is uniformly bounded was considered in , we only consider the case is not uniformly bounded. By (18) and inequalities (), , as , we have
By (5), there exists a constant such that
Set , by (21) and (22) we have
Let in (23), by (15) and (16) we have
Let . By (19), we similarly get
Combining (24) and (25), we obtain (14) directly. □
Let T be a tree, be a stochastic process indexed by the tree T with state-space S. Denote
will be called the entropy density of . If is a T-indexed Markov chain with state-space S defined by Definition 1, we have by (5)
The convergence of to a constant in a sense ( convergence, convergence in probability, a.e. convergence) is called the Shannon-McMillan theorem or the entropy theorem or the AEP in information theory.
Theorem 2 Let T be an infinite tree with assumption (5) holds. Let , and P be an ergodic stochastic matrix. Denote the unique stationary distribution of P by π. Let be a T-indexed Markov chain with state-space S generated by P. Then, for given integer ,
Let , then
Let be defined as (27), then
Proof The proofs of (28) and (29) are similar to those of Huang and Yang (, Theorem 2 and Corollary 3). Letting in Lemma 1, then
by (29), (30) holds. □
Takacs C: Strong law of large numbers for branching Markov chains. Markov Process. Relat. Fields 2001, 8: 107–116.
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 2008,51(2):195–202. 10.1007/s11425-008-0015-1
Benjamini I, Peres Y: Markov chains indexed by trees. Ann. Probab. 1994, 22: 219–243. 10.1214/aop/1176988857
Berger T, Ye Z: Entropic aspects of random fields on trees. IEEE Trans. Inf. Theory 1990, 36: 1006–1018. 10.1109/18.57200
Ye Z, Berger T: Ergodic, regular and asymptotic equipartition property of random fields on trees. J. Comb. Inf. Syst. Sci. 1996, 21: 157–184.
Pemantle R: Automorphism invariant measure on trees. Ann. Probab. 1992, 20: 1549–1566. 10.1214/aop/1176989706
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 2001,21(4):495–502.
Yang WG: Some limit properties for Markov chains indexed by a homogeneous tree. Stat. Probab. Lett. 2003, 65: 241–250. 10.1016/j.spl.2003.04.001
Yang WG, Ye Z: The asymptotic equipartition property for nonhomogeneous Markov chains indexed by a homogeneous tree. IEEE Trans. Inf. Theory 2007,53(9):3275–3280.
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 2007,27(4):829–837. 10.1016/S0252-9602(07)60080-0
This work is supported by the Foundation of Anhui Educational Committee (No. KJ2014A174).
The author declares that they have no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
About this article
Cite this article
Peng, Wc. A note on the strong law of large numbers for Markov chains indexed by irregular trees. J Inequal Appl 2014, 244 (2014). https://doi.org/10.1186/1029-242X-2014-244
- Markov chain
- strong law of large numbers