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October, 1975 The First Birth Problem for an Age-dependent Branching Process
J. F. C. Kingman
Ann. Probab. 3(5): 790-801 (October, 1975). DOI: 10.1214/aop/1176996266

Abstract

If $B_n$ denotes the time of the first birth in the $n$th generation of an age-dependent branching process of Crump-Mode type, then under a weak condition there is a constant $\gamma$ such that $B_n/n \rightarrow \gamma$ as $n \rightarrow \infty$, almost surely on the event of ultimate survival. This strengthens a result of Hammersley, who proved convergence in probability for the more special Bellman-Harris process. The proof depends on a class of martingales which arise from a `collective marks' argument.

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J. F. C. Kingman. "The First Birth Problem for an Age-dependent Branching Process." Ann. Probab. 3 (5) 790 - 801, October, 1975. https://doi.org/10.1214/aop/1176996266

Information

Published: October, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0325.60079
MathSciNet: MR400438
Digital Object Identifier: 10.1214/aop/1176996266

Subjects:
Primary: 60J80
Secondary: 60F10 , 60F15 , 60G45

Keywords: Age-dependent branching processes , Almost sure convergence , Martingales , Subadditive processes

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 5 • October, 1975
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