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On the class of 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, 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.
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Abstract

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

Keywords

Controlled branching processesextinction probabilitylimit theorems

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 804 - 815
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research supported by the Plan Nacional de Investigación Cientifíca, Desarrollo e Innovación Tecnológica, grant BFM 2000-0356.

References

Bagley, J. H. (1986). On almost sure convergence of controlled branching processes. J. Appl. Prob. 23, 827–831.Google Scholar
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. In Branching Processes (Lecture Notes Statist. 99), ed. Heyde, C. C., Springer, Berlin, pp. 14–21.Google Scholar
Feller, W. (1968). Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Holzheimer, J. (1984). ϕ-branching processes in a random environment. Zastos. Mat. 18, 351–358.Google Scholar
Molina, M., González, M., and Mota, M. (1998). Some theoretical results about superadditive controlled Galton–Watson branching processes. In Proc. Prague Stoch.’98, Vol. 1, eds Husková, M., Lachout, P. and Visek, J. A., Union of Czech Mathematicians and Physicists, Prague, pp. 413–418.Google Scholar
Sevast'yanov, B. A. and Zubkov, A. M. (1974). Controlled branching processes. Theory Prob. Appl. 19, 14–24.CrossRefGoogle Scholar
Stromberg, K. R. (1981). An Introduction to Real Analysis. Wadworth and Brooks, Belmont, CA.Google Scholar
Yanev, N. M. (1976). Conditions for degeneracy of φ-branching processes with random φ. Theory Prob. Appl. 20, 421–428.Google Scholar
Yanev, G. P., and Yanev, N. M. (1989). Conditions for extinction of controlled branching processes. In Mathematics and Education in Mathematics, ed. Gerov, G., Bulgarian Akad. Nauk., Sofia, pp. 550–555.Google Scholar
Yanev, G. P., and Yanev, N. M. (1990). Extinction of controlled branching processes in random environments. Math. Balkanica 4, 368–380.Google Scholar
Zubkov, A. M. (1974). Analogies between Galton–Watson processes and φ-branching processes. Theory Prob. Appl. 19, 309–331.Google Scholar