Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Continuous-state branching processes, extremal processes and super-individuals

Clément Foucart and Chunhua Ma

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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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1061-1086.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820842

Digital Object Identifier
doi:10.1214/18-AIHP909

Mathematical Reviews number (MathSciNet)
MR3949964

Zentralblatt MATH identifier
07097342

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G70: Extreme value theory; extremal processes 60G55: Point processes

Keywords
Continuous-state branching process Subordinator Extremal process Infinite mean Infinite variation Super-exponential growth Grey martingale Non-linear renormalisation

Citation

Foucart, Clément; Ma, Chunhua. Continuous-state branching processes, extremal processes and super-individuals. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1061--1086. doi:10.1214/18-AIHP909. https://projecteuclid.org/euclid.aihp/1557820842


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