Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T20:33:21.620Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential ergodicity for single-birth processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Yong-Hua Mao  and
Yu-Hui Zhang
Yong-Hua Mao*
Affiliation:
Beijing Normal University
Yu-Hui Zhang*
Affiliation:
Beijing Normal University
*
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China
Get access Rights & Permissions [Opens in a new window]

Abstract

An explicit, computable, and sufficient condition for exponential ergodicity of single-birth processes is presented. The corresponding criterion for birth–death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods.

Keywords

Exponentially ergodicsingle-birth processbirth–death process

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1022 - 1032
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

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Andreev, D. B. et al.(2002). On ergodicity and stability estimates for some nonhomogeneous Markov chains. J. Math. Sci. 112, 4111–4118.Google Scholar
Chen, A.-Y. (2002). Ergodicity and stability of generalised Markov branching processes with resurrection. J. Appl. Prob. 39, 786–803.Google Scholar
Chen, A.-Y. (2002). Uniqueness and extinction properties of generalised Markov branching processes. J. Math. Anal. Appl. 274, 482–494.Google Scholar
Chen, M.-F. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.Google Scholar
Chen, M.-F. (1999). Single birth processes. Chinese Ann. Math. B 20, 77–82.Google Scholar
Chen, M.-F. (2000). Explicit bounds of the first eigenvalue. Sci. China A 43, 1051–1059.Google Scholar
Chen, M.-F. (2001). Explicit criteria for several types of ergodicity. Chinese J. Appl. Statist. 17, 113–120.Google Scholar
Chen, M.-F. (2004). Eigenvalues, Inequalities and Ergodic Theory. Springer, New York.Google Scholar
Chen, R.-R. (1997). An extended class of time-continuous branching processes. J. Appl. Prob. 34, 14–23.CrossRefGoogle Scholar
Kijima, M. (1992). Evaluation of the decay parameter for some specialized birth–death processes. J. Appl. Prob. 29, 781–791.Google Scholar
Kijima, M. (1993). Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous time. J. Appl. Prob. 30, 509–517.CrossRefGoogle Scholar
Lin, X., and Zhang, H.-J. (2002). The convergence property of branching process. Preprint (in Chinese).Google Scholar
Van Doorn, E. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv. Appl. Prob. 17, 504–530.Google Scholar
Wang, Z.-K. (1996). Introduction to Stochastic Processes, Vol. 2. Beijing Normal University Press (in Chinese).Google Scholar
Yan, S.-J., and Chen, M.-F. (1986). Multidimensional Q-processes. Chinese Ann. Math. B 7, 90–110.Google Scholar
Zeifman, A. I. (1991). Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob. 28, 268–277.Google Scholar
Zeifman, A. I. (1995). Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Process. Appl. 59, 157–173.Google Scholar
Zhang, J. K. (1984). On the generalized birth and death processes. I. Acta Math. Sci. 4, 241–259.Google Scholar
Zhang, Y.-H. (2001). Strong ergodicity for single-birth processes. J. Appl. Prob. 38, 270–277.CrossRefGoogle Scholar