Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T17:05:26.319Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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

[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