On perturbation bounds for continuous-time Markov chains

2014 ◽  
Vol 88 ◽  
pp. 66-72 ◽  
Author(s):  
A.I. Zeifman ◽  
V.Yu. Korolev
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains

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.

Stability and exponential convergence of continuous-time Markov chains

2003 ◽  
Vol 40 (04) ◽  
pp. 970-979 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Exponential Convergence ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
The Relationship

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

The spectral gap and perturbation bounds for reversible continuous-time Markov chains

2004 ◽  
Vol 41 (4) ◽  
pp. 1219-1222 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Spectral Gap ◽  
Complete Information ◽  
Stationary States ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
Distribution Vector

We show that, for reversible continuous-time Markov chains, the closeness of the nonzero eigenvalues of the generator to zero provides complete information about the sensitivity of the distribution vector to perturbations of the generator. Our results hold for both the transient and the stationary states.

Stability and exponential convergence of continuous-time Markov chains

2003 ◽  
Vol 40 (4) ◽  
pp. 970-979 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Exponential Convergence ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
The Relationship

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

The spectral gap and perturbation bounds for reversible continuous-time Markov chains

2004 ◽  
Vol 41 (04) ◽  
pp. 1219-1222 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Spectral Gap ◽  
Complete Information ◽  
Stationary States ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
Distribution Vector

We show that, for reversible continuous-time Markov chains, the closeness of the nonzero eigenvalues of the generator to zero provides complete information about the sensitivity of the distribution vector to perturbations of the generator. Our results hold for both the transient and the stationary states.

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