Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T20:00:44.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of critical controlled branching processes with random control functions

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

M. González ,
M. Molina  and
I. del Puerto
M. González*
Affiliation:
Universidad de Extremadura
M. Molina*
Affiliation:
Universidad de Extremadura
I. del Puerto*
Affiliation:
Universidad de Extremadura
*
∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain.
∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain.
∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain.
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.

In this paper, we investigate the asymptotic behaviour of controlled branching processes with random control functions. In a critical case, we establish sufficient conditions for both their almost-sure extinction and for their nonextinction with a positive probability. For some suitably chosen norming constants, we also determine different kinds of limiting behaviour for this class of processes.

Keywords

Branching processcontrolled processextinction probabilitylimiting behaviour

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 463 - 477
Copyright
© Applied Probability Trust 2005 

References

Bruss, F. T. (1980). A counterpart of the Borel–Cantelli lemma. J. Appl. Prob. 17, 1094–1101.Google Scholar
Chow, Y. S. and Teicher, H. (1997). Probability Theory. Independence, Interchangeability, Martingales, 3rd edn. Springer, New York.Google Scholar
Dion, J.-P. and Essebbar, B. (1995). On the statistics of controlled branching processes. In Branching Processes (Lecture Notes Statist. 99), ed. Heyde, C. C., Springer, New York, pp. 14–21.CrossRefGoogle Scholar
González, M., Molina, M. and del Puerto, I. (2002). On the controlled Galton–Watson process with random control function. J. Appl. Prob. 39, 804–815.Google Scholar
González, M., Molina, M. and del Puerto, I. (2003). On the geometric growth in controlled branching processes with random control function. J. Appl. Prob. 40, 995–1006.Google Scholar
González, M., Molina, M. and del Puerto, I. (2004). Limiting distribution for subcritical controlled branching processes with random control function. Statist. Prob. Lett. 67, 277–284.Google Scholar
González, M., Molina, M. and del Puerto, I. (2005). On L2-convergence of controlled branching processes with random control function. Bernoulli 11, 37–46.Google Scholar
Höpfner, R. (1985). On some classes of population-size-dependent Galton–Watson processes. J. Appl. Prob. 22, 25–36.Google Scholar
Höpfner, R. (1986). Some results on population-size-dependent Galton–Watson processes. J. Appl. Prob. 23, 297–306.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Keller, G., Kersting, G. and Rösler, U. (1987). On the asymptotic behaviour of discrete time stochastic growth processes. Ann. Prob. 15, 305–343.Google Scholar
Kersting, G. (1986). On recurrence and transience of growth models. J. Appl. Prob. 23, 614–625.Google Scholar
Kersting, G. (1992). Asymptotic Γ-distribution for stochastic difference equations. Stoch. Process. Appl. 40, 15–28.Google Scholar
Klebaner, F. (1989). Stochastic difference equations and generalized gamma distributions. Ann. Prob. 17, 178–188.Google Scholar
Nakagawa, T. (1994). The Lalpha (1<α≤2) convergence of a controlled branching process in a random environment. Bull. Gen. Ed. Dokkyo Univ. School Medicine 17, 17–24.Google Scholar
Yanev, G. P. and Yanev, N. M. (1995). Critical branching process with random migration. In Branching Processes (Lecture Notes Statist. 99), ed. Heyde, C. C., Springer, New York, pp. 36–46.Google Scholar
Yanev, G. P. and Yanev, N. M. (2004). A critical branching process with stationary-limiting distribution. Stoch. Anal. Appl. 22, 721–738.Google Scholar
Yanev, G. P., Mitov, K. V. and Yanev, N. M. (2003). Critical branching regenerative process with random migration. J. Appl. Statist. Sci. 12, 41–54.Google Scholar
Yanev, N. M. (1976). Conditions for degeneracy of ɸ-branching processes with random ɸ. Theory Prob. Appl. 20, 421–428.Google Scholar