Journal of Applied Probability

Domain of attraction of the quasistationary distribution for birth-and-death processes

Hanjun Zhang and Yixia Zhu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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, {ajC)}, 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\→∞M(X(t)=jT>t)= ajC), 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<∞.

Article information

Source
J. Appl. Probab., Volume 50, Number 1 (2013), 114-126.

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1363784428

Digital Object Identifier
doi:10.1239/jap/1363784428

Mathematical Reviews number (MathSciNet)
MR3076776

Zentralblatt MATH identifier
1282.60088

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Domain of attraction quasistationary distribution birth-and-death process orthogonal polynomial duality

Citation

Zhang, Hanjun; Zhu, Yixia. Domain of attraction of the quasistationary distribution for birth-and-death processes. J. Appl. Probab. 50 (2013), no. 1, 114--126. doi:10.1239/jap/1363784428. https://projecteuclid.org/euclid.jap/1363784428


Export citation

References

  • Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.
  • Cavender, J. A. (1978). Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570–586.
  • Chen, A. and Zhang, H. (1999). Existence, uniqueness, and constructions for stochastically monotone Q-processes. Southeast Asian Bull. Math. 23, 559–583.
  • Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pure. Appl. 38, 301–345.
  • 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.
  • 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.
  • Karlin, S. and McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.
  • Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London. Math. Soc. 13, 337–358.
  • Pakes, A. G. (1995). Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state. Adv. Appl. Prob. 27, 120–145.
  • Van Doorn, E. A. (1986). On orthogonal polynomials with positive zeros and the associated kernel polynomials. J. Math. Anal. Appl. 113, 441–450.
  • Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700.
  • 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.
  • 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.