Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of coalescent processes near time zero

Batı Şengül

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Abstract

In this paper we obtain scaling limits of $\Lambda$-coalescents near time zero under a regularly varying assumption. In particular this covers the case of Kingman’s coalescent and beta coalescents. The limiting processes are coalescents with infinite mass, obtained geometrically as tangent cones of Evans metric space associated with the coalescent. In the case of Kingman’s coalescent we are able to obtain a simple construction of the limiting space using a two-sided Brownian motion.

Résumé

Nous obtenons des limites d’échelle de $\Lambda$-coalescents en temps zéro sous une hypothèse de variation régulière. Cette hypothèse inclut notamment le coalescent de Kingman ainsi que la famille des Beta-coalescents. Les processus limites sont des processus de coalescence avec masse infinie, construits de manière géométrique comme cônes tangents de l’espace métrique de Evans. Dans le cas particulier du coalescent de Kingman une construction simple du processus limite est donnée à partir d’un mouvement brownien bidirectionnel.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 616-640.

Dates
Received: 17 March 2014
Revised: 1 October 2015
Accepted: 5 November 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897738

Digital Object Identifier
doi:10.1214/15-AIHP727

Mathematical Reviews number (MathSciNet)
MR3634267

Zentralblatt MATH identifier
1367.60030

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J99: None of the above, but in this section 60F99: None of the above, but in this section

Keywords
Regularly varying coalescents Small time asymptotics Scaling limits Random metric space Tangent cones Gromov–Hausdorff convergence

Citation

Şengül, Batı. Scaling limits of coalescent processes near time zero. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 616--640. doi:10.1214/15-AIHP727. https://projecteuclid.org/euclid.aihp/1491897738


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