Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case

Amine Asselah, Pablo A. Ferrari, Pablo Groisman, and Matthieu Jonckheere

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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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 647-668.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367888

Digital Object Identifier
doi:10.1214/14-AIHP635

Mathematical Reviews number (MathSciNet)
MR3498004

Zentralblatt MATH identifier
1342.60145

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces

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

Citation

Asselah, Amine; Ferrari, Pablo A.; Groisman, Pablo; Jonckheere, Matthieu. Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 647--668. doi:10.1214/14-AIHP635. https://projecteuclid.org/euclid.aihp/1462367888


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