A Strong Law of Large Numbers for Markov Chain Fields on a Special Non-homogeneous Tree

2009 ◽  
Author(s):  
Jin Shaohua ◽  
Zhong Yanyan ◽  
Wan Yanping ◽  
Chen Xiuyin
Keyword(s):  
Markov Chain ◽  
Law Of Large Numbers ◽  
Homogeneous Tree ◽  
Strong Law ◽  
Large Numbers

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.

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 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.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (3) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (03) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

On record and inter-record times for a sequence of random variables defined on a Markov chain

1975 ◽  
Vol 7 (01) ◽  
pp. 195-214 ◽  
Author(s):  
Gary Lee Guthrie ◽  
Paul T. Holmes
Keyword(s):  
Markov Chain ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Finite Markov Chain ◽  
Record Times ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Continuous Parameter ◽  
Large Numbers

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

On record and inter-record times for a sequence of random variables defined on a Markov chain

1975 ◽  
Vol 7 (1) ◽  
pp. 195-214 ◽  
Author(s):  
Gary Lee Guthrie ◽  
Paul T. Holmes
Keyword(s):  
Markov Chain ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Finite Markov Chain ◽  
Record Times ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Continuous Parameter ◽  
Large Numbers

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

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