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.

scholarly journals Limit theorems for asymptotic circular mth-order Markov chains indexed by an m-rooted homogeneous tree

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1817-1832
Author(s):  
Huilin Huang ◽  
Weiguo Yang
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Homogeneous Tree ◽  
Strong Law ◽  
Nonhomogeneous Markov Chain ◽  
Asymptotic Equipartition Property ◽  
Large Numbers ◽  
Definition Of ◽  
Rooted Homogeneous Tree

In this paper, we give the definition of an asymptotic circularmth-order Markov chain indexed by an m rooted homogeneous tree. By applying the limit property for a sequence of multi-variables functions of a nonhomogeneous Markov chain indexed by such tree, we estabish the strong law of large numbers and the asymptotic equipartition property (AEP) for asymptotic circular mth-order finite Markov chains indexed by this homogeneous tree. As a corollary, we can obtain the strong law of large numbers and AEP about the mth-order finite nonhomogeneous Markov chain indexed by the m rooted homogeneous tree.

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.

scholarly journals Strong Law of Large Numbers of the Offspring Empirical Measure for Markov Chains Indexed by Homogeneous Tree

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Huilin Huang
Keyword(s):  
Markov Chains ◽  
Law Of Large Numbers ◽  
Empirical Measure ◽  
Homogeneous Tree ◽  
Almost Everywhere Convergence ◽  
Strong Law ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere ◽  
Large Numbers

We study the limit law of the offspring empirical measure and for Markov chains indexed by homogeneous tree with almost everywhere convergence. Then we prove a Shannon-McMillan theorem with the convergence almost everywhere.

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 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 Cayley Trees

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Huilin Huang
Keyword(s):  
Markov Chains ◽  
Law Of Large Numbers ◽  
Hidden Markov ◽  
Sample Entropy ◽  
Entropy Rate ◽  
Strong Limit ◽  
Hidden Markov Chains ◽  
Strong Law ◽  
Cayley Trees ◽  
Large Numbers

We extend the idea of hidden Markov chains on lines to the situation of hidden Markov chains indexed by Cayley trees. Then, we study the strong law of large numbers for hidden Markov chains indexed by Cayley trees. As a corollary, we get the strong limit law of the conditional sample entropy rate.

Sign in / Sign up

Export Citation Format

Share Document