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The eigentime identity for continuous-time ergodic Markov chains

Part of: Markov processes General theory of linear operators

Published online by Cambridge University Press:  14 July 2016

Yong-Hua Mao
Yong-Hua Mao*
Affiliation:
Beijing Normal University
*
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China. Email address: maoyh@bnu.edu.cn
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Abstract

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 ≤ λ2 ≤ ···} be the whole spectrum of L in L2. Then ∑ n≥1 λ n -1 < ∞ implies the existence of the spectral gap α of L in L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑ n≥1 λ n -1 < ∞ if and only if α > 0.

Keywords

Markov chaineigentime identitydeviation matrixhitting timestrong ergodicityeigenvalue

MSC classification

Primary: 60J75: Jump processes 47A75: Eigenvalue problems
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1071 - 1080
Copyright
Copyright © Applied Probability Trust 2004 

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References

Aldous, D. J., and Fill, J. A. (2003). Reversible Markov Chains and Random Walks on Graphs. In preparation. Drafts of chapters available at http://www.stat.berkeley.edu/users/aldous/.Google Scholar
Anderson, W. (1991). Continuous-Time Markov Chains. Springer, New York.CrossRefGoogle Scholar
Chen, M. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.CrossRefGoogle Scholar
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
Coolen-Schrijner, P., and van Doorn, E. A. (2002). The deviation matrix of a continuous-time Markov chain. Prob. Eng. Inf. Sci. 16, 351–366.CrossRefGoogle Scholar
Hou, Z. T., and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.Google Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modelling. Chapman and Hall, London.CrossRefGoogle Scholar
Mao, Y.-H. (2002). Strong ergodicity for Markov processes by coupling methods. J. Appl. Prob. 39, 839–852.CrossRefGoogle Scholar
Mao, Y.-H. (2003). Algebraic convergence for discrete-time Markov chains. Sci. China A 46, 621–630.CrossRefGoogle Scholar
Mao, Y.-H. (2004). Ergodic degrees for continuous-time Markov chains. Sci. China A 47, 161–174.CrossRefGoogle Scholar
Reed, M., and Simon, B. (1978). Methods of Modern Mathematical Physics, Vols I, II. Academic Press, New York.Google Scholar
Saloff-Coste, L. (1994). Precise estimates on the rate at which certain diffusions tend to equilibrium. Math. Z. 217, 641–677.CrossRefGoogle Scholar
Syski, R. (1978). Ergodic potential. Stoch. Process. Appl. 7, 311–336.CrossRefGoogle Scholar
Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Prob. Theory Relat. Fields 128, 255–321.CrossRefGoogle Scholar
Zhang, H. J., Lin, X., and Hou, Z. T. (2000). Polynomial uniform convergence for standard transition functions. Chinese Ann. Math. A 21, 351–355 (in Chinese).Google Scholar
Zhang, Y.-H. (2001). Strong ergodicity for single-birth processes. J. Appl. Prob. 38, 270–277.CrossRefGoogle Scholar