Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T23:21:35.015Z Has data issue: false hasContentIssue false
  • English
  • Français

The λ-classification of continuous-time birth-and-death processes

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Andrew G. Hart ,
Servet Martínez  and
Jaime San Martín
Andrew G. Hart*
Affiliation:
Universidad de Chile, Santiago
Servet Martínez*
Affiliation:
Universidad de Chile, Santiago
Jaime San Martín*
Affiliation:
Universidad de Chile, Santiago
*
∗ Postal address: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071 CNRS-UCHILE, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile.
∗ Postal address: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071 CNRS-UCHILE, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile.
∗ Postal address: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071 CNRS-UCHILE, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent.

Keywords

Birth-and-death processdecay parameterλ-theoryabsorptionnonexplosivity

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1111 - 1130
Copyright
Copyright © Applied Probability Trust 2003 

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: an Applications-Oriented Approach. Springer, New York.Google Scholar
Bean, N. G., Pollett, P. K. and Taylor, P. G. (2000). The quasistationary distributions of level-dependent quasi-birth-and-death processes. Stoch. Models 16, 511–541.Google Scholar
Collet, P., Martínez, S. and San Martín, J. (1995). Asymptotic laws for one dimensional diffusions conditioned to nonabsorption. Ann. Prob. 23, 1300–1314.CrossRefGoogle Scholar
Elmes, S., Pollett, P. K. and Walker, D. M. (2000). Further results on the relationship between μ-invariant measures and quasistationary distributions for continuous-time Markov chains. Math. Comput. Modelling 31, 107–113.Google Scholar
Ferrari, P. A., Kesten, H. and Martínez, S. (1996). R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata. Ann. Appl. Prob. 6, 577–616.Google Scholar
Hart, A. G. (1997). Quasistationary distributions for continuous-time Markov chains. Doctoral Thesis, University of Queensland.Google Scholar
Hart, A. G. and Pollett, P. K. (1996). Direct analytical methods for determining quasistationary distributions for continuous-time Markov chains. In Proc. 1995 Athens Conf. Appl. Prob. Time Series Anal., Vol. 1, Applied Probability (Lecture Notes Statist. 114), eds Heyde, C., Prohorov, Y. V., Pyke, R. and Rachev, S., Springer, New York, pp. 116–126.Google Scholar
Jacka, S. D. and Roberts, G. O. (1996). Weak convergence of conditioned processes on a countable state space. J. Appl. Prob. 32, 902–916.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.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.Google Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.Google Scholar
Kijima, M., Nair, M. G., Pollett, P. K. and van Doorn, E. A. (1997). Limiting conditional distributions for birth–death processes. Adv. Appl. Prob. 29, 185–204.CrossRefGoogle Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337–358.Google Scholar
Martínez, S. and San Martín, J. (2003). Classification of killed one-dimensional diffusions. To appear in Ann. Prob. Google Scholar
Pollett, P. K. (1986). On the equivalence of μ-invariant measures for the minimal process and its q-matrix. Stoch. Process. Appl. 22, 203–221.Google Scholar
Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob. 20, 600–621.Google Scholar
Roberts, G. O. and Jacka, S. D. (1994). Weak convergence of conditioned birth and death processes. J. Appl. Prob. 31, 90–100.Google Scholar
Roberts, G. O., Jacka, S. D. and Pollett, P. K. (1997). Non-explosivity of limits of conditioned birth and death processes. J. Appl. Prob. 34, 35–45.CrossRefGoogle Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.CrossRefGoogle Scholar