The asymptotic equipartition property of Markov chains in single infinite Markovian environment on countable state space

Stochastics ◽  
2019 ◽  
Vol 91 (6) ◽  
pp. 945-957 ◽  
Author(s):  
Zhiyan Shi ◽  
Dan Bao ◽  
Yan Fan ◽  
Baihui Wu
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Countable State Space ◽  
Markovian Environment ◽  
Asymptotic Equipartition Property ◽  
Countable State

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 Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 253 ◽  
Author(s):  
Alexander Zeifman ◽  
Victor Korolev ◽  
Yacov Satin
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Countable State Space ◽  
Perturbation Bounds ◽  
Queuing Models ◽  
Continuous Time Markov Chains ◽  
Quantitative Estimates ◽  
Countable State

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

1989 ◽  
Vol 26 (3) ◽  
pp. 643-648 ◽  
Author(s):  
A. I. Zeifman
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

Integral representations of transition probabilities and serial covariances of certain markov chains

1969 ◽  
Vol 6 (03) ◽  
pp. 648-659 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Spectral Measure ◽  
Transition Probabilities ◽  
Integral Representations ◽  
Stationary Processes ◽  
Birth And Death Processes ◽  
Countable State Space ◽  
Completely Monotonic ◽  
Countable State

This note is concerned with integral representations of some transition probabilities of countable state space Markov chains embedded in birth and death processes, of the serial covariances of functions defined on such chains when stationary, and finally the properties of the spectral measure of stationary processes with monotonically decreasing or completely monotonic serial covariances.

scholarly journals Path integrals for continuous-time Markov chains

2002 ◽  
Vol 39 (4) ◽  
pp. 901-904 ◽  
Author(s):  
P. K. Pollett ◽  
V. T. Stefanov
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Path Integral ◽  
Continuous Time ◽  
Path Integrals ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

1989 ◽  
Vol 26 (03) ◽  
pp. 643-648 ◽  
Author(s):  
A. I. Zeifman
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l 1 . Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

scholarly journals Path integrals for continuous-time Markov chains

2002 ◽  
Vol 39 (04) ◽  
pp. 901-904 ◽  
Author(s):  
P. K. Pollett ◽  
V. T. Stefanov
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Path Integral ◽  
Continuous Time ◽  
Path Integrals ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

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