Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T13:02:05.238Z Has data issue: false hasContentIssue false
  • English
  • Français

Domain of Attraction of the Quasistationary Distribution for Birth-and-Death Processes

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Hanjun Zhang  and
Yixia Zhu
Hanjun Zhang*
Affiliation:
Xiangtan University
Yixia Zhu*
Affiliation:
Xiangtan University
*
∗ Postal address: School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P. R. China.
∗ Postal address: School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P. R. China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λn,n≥0, and death coefficients μn,n≥0. If we define A=∑n=1∞ 1/λnπn and S=∑n=1∞ (1/λnπn)∑i=n+1∞ πi, where {πn,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λC>0 and there is precisely one quasistationary distribution, namely, {aj(λC)}, where λC is the decay parameter of {X(t),t≥0} in C={1,2,...} and aj(x)≡μ1-1πjxQj(x), j=1,2,.... In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={mi, i=1,2,...}, we have limt→∞ℙM(X(t)=j∣ T>t)= aj(λC), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

Keywords

Domain of attractionquasistationary distributionbirth-and-death processorthogonal polynomialduality

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 114 - 126
Copyright
Copyright © Applied Probability Trust 2013 

References

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Cavender, J. A. (1978). Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570–586.Google Scholar
Chen, A. and Zhang, H. (1999). Existence, uniqueness, and constructions for stochastically monotone Q-processes. Southeast Asian Bull. Math. 23, 559–583.Google Scholar
Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pure. Appl. 38, 301–345.Google Scholar
Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501–521.Google Scholar
Karlin, S. and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
Karlin, S. and McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London. Math. Soc. 13, 337–358.Google Scholar
Pakes, A. G. (1995). Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state. Adv. Appl. Prob. 27, 120–145.CrossRefGoogle Scholar
Van Doorn, E. A. (1986). On orthogonal polynomials with positive zeros and the associated kernel polynomials. J. Math. Anal. Appl. 113, 441–450.Google Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700.Google Scholar
Zhang, H. and Liu, W. (2012). Domain of attraction of the quasi-stationary distribution for the linear birth and death process. J. Math. Anal. Appl. 385, 677–682.Google Scholar
Zhang, H., Liu, W., Peng, X. and Liu, S. (2012). Domain of attraction of the quasi-stationary distributions for the birth and death process. Preprint.Google Scholar