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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Huilin Huang
Keyword(s):  
Markov Chains ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Hidden Markov ◽  
Strong Limit ◽  
Hidden Markov Chains ◽  
Strong Law ◽  
Large Numbers ◽  
Infinite Tree ◽  
Uniformly Bounded

We study strong limit theorems for hidden Markov chains fields indexed by an infinite tree with uniformly bounded degrees. We mainly establish the strong law of large numbers for hidden Markov chains fields indexed by an infinite tree with uniformly bounded degrees and give the strong limit law of the conditional sample entropy rate.

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.

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.

A Note on External Uniformization for Finite Markov Chains in Continuous Time

1992 ◽  
Vol 6 (1) ◽  
pp. 127-131 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Numerical Experiments ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Strong Law ◽  
Large Numbers ◽  
Finite Markov Chains

An external uniformization technique was developed by Ross [4] to obtain approximations of transition probabilities of finite Markov chains in continuous time. Yoon and Shanthikumar [7] then reported through extensive numerical experiments that this technique performs quite well compared to other existing methods. In this paper, we show that external uniformization results from the strong law of large numbers whose underlying distributions are exponential. Based on this observation, some remarks regarding properties of the approximation are given.

scholarly journals The asymptotic behavior for Markov chains in a finite i.i.d random environment indexed by cayley trees

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 273-283 ◽  
Author(s):  
Huilin Huang
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Homogeneous Tree ◽  
Strong Law ◽  
Cayley Trees ◽  
Large Numbers ◽  
Finite Markov Chains

We firstly define a Markov chain indexed by a homogeneous tree in a finite i.i.d random environment. Then, we prove the strong law of large numbers and Shannon-McMillan theorem for finite Markov chains indexed by a homogeneous tree in the finite i.i.d random environment.

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