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.
Publié le : 1975-10-14
Classification:
Age-dependent branching processes,
almost sure convergence,
subadditive processes,
martingales,
60J80,
60F10,
60F15,
60G45
@article{1176996266,
author = {Kingman, J. F. C.},
title = {The First Birth Problem for an Age-dependent Branching Process},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 790-801},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996266}
}
Kingman, J. F. C. The First Birth Problem for an Age-dependent Branching Process. Ann. Probab., Tome 3 (1975) no. 6, pp. 790-801. http://gdmltest.u-ga.fr/item/1176996266/