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The stable pedigrees of critical branching populations

Published online by Cambridge University Press:  14 July 2016

Olle Nerman
Olle Nerman*
Affiliation:
University of Göteborg
*
∗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

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

Keywords

BRANCHING PROCESSPOPULATION DYNAMICSSTABLE AGE DISTRIBUTIONCELL KINETICS
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 3 , September 1984 , pp. 447 - 463
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

This work has been supported by a grant from the Swedish Natural Science Research Council.

References

[1] 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.CrossRefGoogle Scholar
[2] Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
[3] Green, P. J. (1977) Conditional limit theorems for general branching processes. J. Appl. Prob. 14, 451–463.CrossRefGoogle Scholar
[4] Holte, J. M. (1974) Extinction probability for a critical general branching process. Stoch. Proc. Appl. 2, 303–309.CrossRefGoogle Scholar
[5] Holte, J. M. (1982) Critical multitype branching processes. Ann. Prob. 10, 482–495.Google Scholar
[6] Jagers, P. (1974) Convergence of general branching processes and functionals thereof. J. Appl. Prob. 11, 471–478.Google Scholar
[7] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
[8] Jagers, P. (1982) How probable is it to be first-born? and other branching process applications to kinship problems. Math. Biosci. 59, 1–15.CrossRefGoogle Scholar
[9] Kallenberg, O. (1977) Stability of critical cluster fields. Math. Nachr. 77, 7–43.CrossRefGoogle Scholar
[10] Nerman, O. (1981) On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
[11] Nerman, O. and Jagers, P. (1984) The stable doubly infinite pedigree process of supercritical branching populations. Z. Wahrscheinlichkeitsth. 65, 445–460.Google Scholar