Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T19:25:50.442Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Continuous-State Population-Size-Dependent Branching Process and Its Extinction

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Yuqiang Li
Yuqiang Li*
Affiliation:
Beijing Normal University
*
∗ Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China. Email address: y_q_li@163.com
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.

A continuous-state population-size-dependent branching process {Xt} is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population Xt is discussed, and the limit distribution of Xt / t obtained when Xt tends to infinity.

Keywords

Population-size-dependent branching processfinite-dimensional distributionconvergenceextinctionlimit distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 195 - 207
Copyright
© Applied Probability Trust 2006 

References

Athreya, K. B. and Ney, P. (1972). Branching Processes. Springer, Berlin.Google Scholar
Ethier, S. N. and Kurtz, T. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Höpfner, R. (1985). On some classes of population-size-dependent Galton–Watson processes. J. Appl. Prob. 22, 25–36.Google Scholar
Jagers, P. (1996). Population-size-dependent branching processes. J. Appl. Math. Stoch. Analysis 9, 449–457.Google Scholar
Jagers, P. and Klebaner, F. C. (2000). Population-size-dependent and age-dependent branching processes. Stoch. Process. Appl. 87, 235–254.Google Scholar
Jiřina, M. (1960). Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292–313.CrossRefGoogle Scholar
Klebaner, F. C. (1983). Population-size-dependent branching processes with linear rate of growth. J. Appl. Prob. 20, 242–250.Google Scholar
Klebaner, F. C. (1984). Geometric rate of growth in population-size-dependent branching processes. J. Appl. Prob. 21, 40–49.Google Scholar
Klebaner, F. C. (1984). On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.Google Scholar
Klebaner, F. C. (1985). A limit theorem for population-size-dependent branching processes. J. Appl. Prob. 22, 48–57.Google Scholar
Klebaner, F. C. (1993). Population-size-dependent branching processes with a threshold. Stoch. Process. Appl. 46, 115–127.Google Scholar
Labkovskii, V. (1972). A limit theorem for generalized branching random processes depending on the size of the population. Theory Prob. Appl. 17, 72–85.Google Scholar
Li, Z. H. (1991). Integral representations of continuous functions. Chinese Sci. Bull. 36, 979–983.Google Scholar
Li, Z. H. (1992). Measure-valued branching processes with immigration. Stoch. Process. Appl. 43, 249–264.Google Scholar
Lu, Z. W. and Jagers, P. (2004). A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching process with independent and identically distributed random environments. J. Appl. Prob. 41, 176–186.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Seneta, E. and Vere-Jones, D. (1969). On a problem of M. Jiřina concerning continuous state branching processes. Czech. Math. J. 19, 277–283.Google Scholar
Wang, H. X. (1999). Extinction of population-size-dependent branching processes in random environments. J. Appl. Prob. 36, 146–154.Google Scholar
Wang, H. X. and Dai, Y. (1998). Population-size-dependent branching processes in Markovian random environments. Chinese Sci. Bull. 43, 635–638.Google Scholar