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The growth and composition of branching populations

Published online by Cambridge University Press:  01 July 2016

Peter Jagers  and
Olle Nerman
Peter Jagers*
Affiliation:
Chalmers University of Technology and the University of Göteborg
Olle Nerman*
Affiliation:
Chalmers University of Technology and the University of Göteborg
*
∗Postal address: Mathematical Statistics, Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96 Göteborg, Sweden.
∗Postal address: Mathematical Statistics, Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96 Göteborg, Sweden.
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Abstract

A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R+, interpreted as the individuals' ages. Such a population can be measured or counted in many different ways: those born, those alive or in some sub-phase of life, for example. Special choices of reproduction point process and counting yield the classical Galton–Watson or Bellman–Harris process. This reasonably self-contained survey paper discusses the exponential growth of such populations, in the supercritical case, and the asymptotic stability of composition according to very general ways of counting. The outcome is a strict definition of a stable population in exponential growth, as a probability space, a margin of which is the well-known stable age distribution.

Keywords

BRANCHING PROCESSSTABLE POPULATION THEORYBALANCED EXPONENTIAL GROWTH
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 2 , June 1984 , pp. 221 - 259
Copyright
Copyright © Applied Probability Trust 1984 

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References

[1] Asmussen, S. and Hering, H. (1983) Branching Process. Birkhauser, Boston.Google Scholar
[2] Athreya, K. B. and Kaplan, N. (1976) Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process. Ann. Prob. 4, 38–50.Google Scholar
[3] Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[4] Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
[5] Buhler, W. J. (1970) The distribution of generations and other aspects of the family structure of branching processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 463–480.Google Scholar
[6] Doney, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
[7] Doney, R. A. (1976) On single and multitype general age-dependent branching processes. J. Appl. Prob. 13, 239–246.CrossRefGoogle Scholar
[8] Hall, P. and Heyde, C. C. (1983) Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
[9] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[10] Hoppe, F. M. and Seneta, E. (1978) Analytical methods for discrete branching processes. In Joffe, A. and Ney, P. [13].Google Scholar
[11] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, Chichester.Google Scholar
[12] Jagers, P. (1982) How probable is it to be firstborn? and other branching process applications to kinship problems. Math. Biosci. 59, 1–15.CrossRefGoogle Scholar
[13] Joffe, A. and Ney, P., (Eds.) (1978) Branching Processes. Dekker, New York.Google Scholar
[14] Joffe, A. and Waugh, W. A. O'N. (1982) Exact distributions of kin numbers in a Galton-Watson process. J. Appl. Prob. 19, 767–775.Google Scholar
[15] Kuczek, T. (1982) On the convergence of the empiric age distribution for one dimensional supercritical age dependent branching processes. Ann. Prob. 10, 252–258.Google Scholar
[16] Laredo, C. and Rouault, A. (1983) Grandes déviations, dynamique de populations et phénomènes Malthusiens. Ann. Inst. H. Poincaré B 19, 323–350.Google Scholar
[17] Mode, C. J. (1971) Multitype Branching Processes. Elsevier, New York.Google Scholar
[18] Nerman, O. (1981) On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
[19] Nerman, O. and Jagers, P. (1983) The stable doubly infinite pedigree process of supercritical branching processes. Z. Wahrscheinlichkeitsth. Google Scholar
[20] Rouault, A. (1981) Lois empiriques dans les processus de branchement spatiaux homogénes supercritiques. C.R. Acad. Sci. Paris 292(1), 933–936.Google Scholar
[21] Sevast'Yanov, B. A. (1971) Vetvyašciesya Processy. Mir, Moscow. German translation: Verzweigungsprozesse, Akademie-Verlag, Berlin (1974).Google Scholar