We consider a supercritical general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting process of such population is a known as binary homogeneous Crump-Jargers-Mode process. It is known that such processes converges almost surely when correctly renormalized. In this paper, we study the error of this convergence. To this end, we use classical renewal theory and recent works [A. Lambert, Ann. Probab. 38 (2010) 348–395]. on this model to obtain the moments of the error. Then, we can precisely study the asymptotic behaviour of these moments thanks to Lévy processes theory. These results in conjunction with a new decomposition of the splitting trees allow us to obtain a central limit theorem.
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DOI : 10.1051/ps/2016029
Mots-clés : Branching process, splitting tree, Crump–Mode–Jagers process, linear birth–death process, Lévy processes, scale function, Central Limit Theorem
@article{PS_2017__21__113_0,
author = {Henry, Beno{\^\i}t},
title = {Central limit theorem for supercritical binary homogeneous {Crump-Mode-Jagers} processes},
journal = {ESAIM: Probability and Statistics},
pages = {113--137},
publisher = {EDP-Sciences},
volume = {21},
year = {2017},
doi = {10.1051/ps/2016029},
mrnumber = {3716122},
zbl = {1394.60017},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2016029/}
}
TY - JOUR AU - Henry, Benoît TI - Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes JO - ESAIM: Probability and Statistics PY - 2017 SP - 113 EP - 137 VL - 21 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2016029/ DO - 10.1051/ps/2016029 LA - en ID - PS_2017__21__113_0 ER -
%0 Journal Article %A Henry, Benoît %T Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes %J ESAIM: Probability and Statistics %D 2017 %P 113-137 %V 21 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2016029/ %R 10.1051/ps/2016029 %G en %F PS_2017__21__113_0
Henry, Benoît. Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 113-137. doi : 10.1051/ps/2016029. http://www.numdam.org/articles/10.1051/ps/2016029/
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