The Annals of Probability

The First Birth Problem for an Age-dependent Branching Process

J. F. C. Kingman

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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.

Article information

Source
Ann. Probab. Volume 3, Number 5 (1975), 790-801.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996266

Digital Object Identifier
doi:10.1214/aop/1176996266

Mathematical Reviews number (MathSciNet)
MR400438

Zentralblatt MATH identifier
0325.60079

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations 60F15: Strong theorems 60G45

Keywords
Age-dependent branching processes almost sure convergence subadditive processes martingales

Citation

Kingman, J. F. C. The First Birth Problem for an Age-dependent Branching Process. Ann. Probab. 3 (1975), no. 5, 790--801. doi:10.1214/aop/1176996266. https://projecteuclid.org/euclid.aop/1176996266


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