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May 2016 Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case
Amine Asselah, Pablo A. Ferrari, Pablo Groisman, Matthieu Jonckheere
Ann. Inst. H. Poincaré Probab. Statist. 52(2): 647-668 (May 2016). DOI: 10.1214/14-AIHP635

Abstract

Consider $N$ particles moving independently, each one according to a subcritical continuous-time Galton–Watson process unless it hits $0$, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming–Viot process. We show that for each $N$ there exists a unique invariant measure for the Fleming–Viot process, and that its stationary empirical distribution converges, as $N$ goes to infinity, to the minimal quasi-stationary distribution of the Galton–Watson process conditioned on non-extinction.

Nous considérons $N$ particules indépendantes. Chaque particule suit l’évolution d’un processus de Galton–Watson sous-critique jusqu’au moment où elle touche $0$. À cet instant, cette particule choisit uniformément au hasard la position d’une des autres particules et y saute. Ce processus est appelé Fleming–Viot. Nous montrons que pour chaque entier $N$, il existe une unique mesure invariante pour le processus de Fleming–Viot, et que la mesure empirique stationnaire converge vers la loi quasi-stationnaire minimale d’un processus de Galton–Watson conditionné à ne pas mourir.

Citation

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Amine Asselah. Pablo A. Ferrari. Pablo Groisman. Matthieu Jonckheere. "Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case." Ann. Inst. H. Poincaré Probab. Statist. 52 (2) 647 - 668, May 2016. https://doi.org/10.1214/14-AIHP635

Information

Received: 5 September 2013; Revised: 30 June 2014; Accepted: 16 July 2014; Published: May 2016
First available in Project Euclid: 4 May 2016

zbMATH: 1342.60145
MathSciNet: MR3498004
Digital Object Identifier: 10.1214/14-AIHP635

Subjects:
Primary: 60K35
Secondary: 60J25

Keywords: Fleming–Viot processes , Galton–Watson processes , Quasi-stationary distributions , Selection principle

Rights: Copyright © 2016 Institut Henri Poincaré

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Vol.52 • No. 2 • May 2016
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