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Limit theorems for point processes generated in a general branching process

Published online by Cambridge University Press:  01 July 2016

Martin Härnqvist
Martin Härnqvist*
Affiliation:
University of Göteborg
*
∗Postal address: Volvo-Data, Department of Applied Statistics, S-405 08, Göteborg, Sweden.
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Abstract

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit.

Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

Keywords

BRANCHING PROCESSSUPERCRITICALPOINT PROCESSSUPERPOSITIONSPOISSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 650 - 668
Copyright
Copyright © Applied Probability Trust 1981 

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References

Athreya, K. B. and Keiding, N. (1977) Estimation theory for continuous-time branching processes. Sankhyā A 39, 101–123.Google Scholar
Biggins, J. D. (1981) Limiting point processes in the branching random walk. Z. Wahrscheinlichkeitsth. 55, 297–303.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chow, Y. S. and Teicher, H. (1978) Probability Theory: Independence, Interchangeability, Martingales. Springer-Verlag, New York.CrossRefGoogle Scholar
Crump, K. S. and Mode, C. J. (1969) A general age-dependent branching process. J. Math. Anal. Appl. 25, 8–17.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Jagers, P. (1969) A general stochastic model for population development. Skand. Aktuarietidskr. 52, 84–103.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Jagers, P. and Nerman, O. (1980) Limit theorems for sums determined by branching and other exponentially growing processes. Dept of Mathematics, Chalmers University of Technology and University of Göteborg.Google Scholar
Kallenberg, O. (1976) Random Measures. Akademie-Verlag, Berlin; Academic Press, London.Google Scholar
Leadbetter, M. R. (1972) On basic results of point process theory. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 449–462.Google Scholar
Nerman, O. (1981) On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. CrossRefGoogle Scholar