The strong law of large numbers and the Shannon–McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree

2011 ◽  
Vol 81 (12) ◽  
pp. 1883-1890 ◽  
Author(s):  
Yan Dong ◽  
Weiguo Yang ◽  
Jianfang Bai
Keyword(s):  
Markov Chains ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Nonhomogeneous Markov Chains

THE SHANNON–MCMILLAN THEOREM FOR MARKOV CHAINS INDEXED BY A CAYLEY TREE IN RANDOM ENVIRONMENT

2017 ◽  
Vol 32 (4) ◽  
pp. 626-639 ◽  
Author(s):  
Zhiyan Shi ◽  
Pingping Zhong ◽  
Yan Fan
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Random Environment ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Countable State Space ◽  
Large Numbers ◽  
Countable State ◽  
Definition Of

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

SOME LIMIT THEOREMS OF DELAYED AVERAGES FOR COUNTABLE NONHOMOGENEOUS MARKOV CHAINS

2019 ◽  
pp. 1-19
Author(s):  
Pingping Zhong ◽  
Weiguo Yang ◽  
Zhiyan Shi ◽  
Yan Zhang
Keyword(s):  
Information Theory ◽  
Markov Chains ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Strong Ergodicity ◽  
Large Numbers ◽  
Nonhomogeneous Markov Chains ◽  
Definition Of ◽  
Bivariate Functions

AbstractThe purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

STRONG LAW OF LARGE NUMBERS FOR MARKOV CHAINS INDEXED BY SPHERICALLY SYMMETRIC TREES

2015 ◽  
Vol 29 (3) ◽  
pp. 473-481 ◽  
Author(s):  
Peng Weicai ◽  
Yang Weiguo ◽  
Shi Zhiyan
Keyword(s):  
Markov Chains ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Spherically Symmetric ◽  
Strong Law ◽  
Large Numbers ◽  
Symmetric Tree

In this paper, we main consider spherically symmetric tree T. First, under the condition lim supn→∞ |T(n)|/|Ln|<∞, we investigate the strong law of large numbers (SLLNs) for T-indexed Markov chains on the nth level of T. Then, combining the Stolz theorem, we obtain the SLLNs on T. Finally, we get Shannon–McMillan theorem for T-indexed Markov chains. The obtained theorems are generalizations of some known results on Cayley tree TC, N and Bethe tree TB, N.

THE ASYMPTOTIC EQUIPARTITION PROPERTY FOR ASYMPTOTIC CIRCULAR MARKOV CHAINS

2010 ◽  
Vol 24 (2) ◽  
pp. 279-288 ◽  
Author(s):  
Pingping Zhong ◽  
Weiguo Yang ◽  
Peipei Liang
Keyword(s):  
Markov Chains ◽  
Law Of Large Numbers ◽  
Strong Limit ◽  
Strong Law ◽  
Asymptotic Equipartition Property ◽  
Large Numbers ◽  
Nonhomogeneous Markov Chains ◽  
Strong Limit Theorem ◽  
Definition Of ◽  
Bivariate Functions

In this article, we study the asymptotic equipartition property (AEP) for asymptotic circular Markov chains. First, the definition of an asymptotic circular Markov chain is introduced. Then by applying the limit property for the bivariate functions of nonhomogeneous Markov chains, the strong limit theorem on the frequencies of occurrence of states for asymptotic circular Markov chains is established. Next, the strong law of large numbers on the frequencies of occurrence of states for asymptotic circular Markov chains is obtained. Finally, we prove the AEP for asymptotic circular Markov chains.

scholarly journals Strong Law of Large Numbers for Countable Markov Chains Indexed by an Infinite Tree with Uniformly Bounded Degree

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Bao Wang ◽  
Weiguo Yang ◽  
Zhiyan Shi ◽  
Qingpei Zang
Keyword(s):  
Markov Chains ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Bounded Degree ◽  
Strong Law ◽  
Large Numbers ◽  
Infinite Tree ◽  
Finite Markov Chains ◽  
Uniformly Bounded

We study the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for countable Markov chains indexed by an infinite tree with uniformly bounded degree, which extends the corresponding results of countable Markov chains indexed by a Cayley tree and generalizes the relative results of finite Markov chains indexed by a uniformly bounded tree.

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