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May 2019 Continuous-state branching processes, extremal processes and super-individuals
Clément Foucart, Chunhua Ma
Ann. Inst. H. Poincaré Probab. Statist. 55(2): 1061-1086 (May 2019). DOI: 10.1214/18-AIHP909

Abstract

The long-term behavior of flows of continuous-state branching processes are characterized through subordinators and extremal processes. The extremal processes arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to $+\infty$ in the supercritical case, and slow down the rate of extinction in the subcritical one.

Les comportements en temps long des flots de processus de branchement en temps et espace continus sont caractérisés par des subordinateurs et des processus extrémaux. Les processus extrémaux apparaissent dans le cas des processus sur-critiques de moyenne infinie et des processus sous-critiques à variation infinie. Les sauts de ces processus extrémaux sont interprétés comme des individus initiaux spécifiques dont les descendances envahissent la population. Ces individus, qui correspondent aux instants de records d’un certain processus ponctuel de Poisson, sont appelés super-individus. Ils augmentent de façon radicale la vitesse de divergence dans le cas sur-critique et diminuent celle d’extinction dans le cas sous-critique.

Citation

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Clément Foucart. Chunhua Ma. "Continuous-state branching processes, extremal processes and super-individuals." Ann. Inst. H. Poincaré Probab. Statist. 55 (2) 1061 - 1086, May 2019. https://doi.org/10.1214/18-AIHP909

Information

Received: 21 November 2016; Revised: 5 December 2017; Accepted: 16 April 2018; Published: May 2019
First available in Project Euclid: 14 May 2019

zbMATH: 07097342
MathSciNet: MR3949964
Digital Object Identifier: 10.1214/18-AIHP909

Subjects:
Primary: 60J80
Secondary: 60G55, 60G70

Rights: Copyright © 2019 Institut Henri Poincaré

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Vol.55 • No. 2 • May 2019
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