Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T15:33:24.630Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometric growth in controlled branching processes with random control function

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, 06071 Badajoz, Spain
∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, 06071 Badajoz, Spain
∗Postal address: Departamento de Matemáticas, Facultad de Ciencias, 06071 Badajoz, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

The limit behaviour of a controlled branching process with random control function is investigated. A necessary condition and a sufficient condition for the geometric growth of such a process are established by considering the L1-convergence. Finally, taking into account the classical X log+X criterion in branching processes, a necessary and sufficient condition is provided.

Keywords

Controlled branching processlimit theoremL1-convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 995 - 1006
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

Bruss, F. T. (1980). A counterpart of the Borel—Cantelli lemma. J. Appl. Prob. 17, 1094–1101.Google Scholar
Dion, J. P., and Essebbar, B. (1995). On the Statistics of Controlled Branching Processes (Lecture Notes Statist. 99). Springer, New York, pp. 14–21.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
González, M., Molina, M., and del Puerto, I. (2002). On the class of controlled branching process with random control functions. J. Appl. Prob. 39, 804–815.CrossRefGoogle Scholar
Klebaner, F. (1984). Geometric rate of growth in population-size-dependent branching processes. J. Appl. Prob. 21, 40–49.Google Scholar
Klebaner, F. (1985). A limit theorem for population-size-dependent branching processes. J. Appl. Prob. 22, 48–57.Google Scholar
Nakagawa, T. (1994). The Lα (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
Sevastýanov, B. A., and Zubkov, A. M. (1974). Controlled branching processes. Theory Prob. Appl. 19, 14–24.CrossRefGoogle Scholar
Yanev, N. M. (1976). Conditions for degeneracy of ϕ-branching processes with random ϕ. Theory Prob. Appl. 20, 421–428.Google Scholar