Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T19:55:26.618Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

A. Yu. Mitrophanov
A. Yu. Mitrophanov*
Affiliation:
Saratov State University
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Markov chainreversibilityeigenvaluespectral gapcondition numberperturbation boundsensitivity analysis

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1219 - 1222
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. J., and Fill, J. A. (2004). Reversible Markov Chains and Random Walks on Graphs. In preparation. Drafts of chapters available at http://www.stat.berkeley.edu/users/aldous/.Google Scholar
Cho, G. E., and Meyer, C. D. (2001). Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra Appl. 335, 137–150.Google Scholar
Meyer, C. D. (1994). Sensitivity of the stationary distribution of a Markov chain. SIAM J. Matrix Anal. Appl. 15, 715–728.Google Scholar
Mitrophanov, A. Yu. (2003). Stability and exponential convergence of continuous-time Markov chains. J. Appl. Prob. 40, 970–979.Google Scholar
Pritchard, G., and Scott, D. J. (2001). Empirical convergence rates for continuous-time Markov chains. J. Appl. Prob. 38, 262–269.Google Scholar
Seneta, E. (1993). Sensitivity of finite Markov chains under perturbation. Statist. Prob. Lett. 17, 163–168.Google Scholar