Volume 21, 2017
|Page(s)||113 - 137|
|Published online||19 October 2017|
Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes
1 TOSCA project-team, INRIA Nancy – Grand Est, IECL – UMR 7502, Nancy-Université, Campus scientifique, B.P. 70239, 54506 Vandœuvre-lès-Nancy cedex, France.
2 IECL – UMR 7502, Nancy-Université, Campus scientifique, B.P. 70239, 54506 Vandœuvre-lès-Nancy cedex, France.
Received: 21 November 2015
Revised: 6 June 2016
Accepted: 6 December 2016
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.
Mathematics Subject Classification: 60J85 / 60G51 / 60K15 / 60F05
Key words: Branching process / splitting tree / Crump–Mode–Jagers process / linear birth–death process / Lévy processes / scale function / Central Limit Theorem
© EDP Sciences, SMAI, 2017
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