Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Decomposition of Lévy trees along their diameter

Thomas Duquesne and Minmin Wang

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Abstract

We study the diameter of Lévy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Lévy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Lévy trees and we prove that it is realized by a unique pair of points. We prove that the law of Lévy trees conditioned to have a fixed diameter $r\in (0,\infty)$ is obtained by glueing at their respective roots two independent size-biased Lévy trees conditioned to have height $r/2$ and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of Lévy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Lévy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) in the Brownian case.

Résumé

Nous étudions le diamètre des arbres de Lévy qui sont des espaces métriques compacts obtenus comme limites d’échelle des arbres de Galton–Watson. Les arbres de Lévy ont été introduits par Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) et ils généralisent le Continuum Random Tree (1991) d’Aldous qui correspond au cas brownien. Nous caractérisons d’abord la loi du diamètre des arbres de Lévy et nous prouvons qu’une unique paire de points le réalise. Nous prouvons ensuite que la loi des arbres de Lévy conditionnés à avoir leur diamètre égal à $r\in{} ]0,\infty[$ est obtenu en collant à leurs racines respectives deux arbres de Lévy indépendants conditionnés chacuns à avoir une hauteur égale à $r/2$, et à réenraciner uniformément au hasard l’arbre obtenu par ce collage ; nous décrivons également en termes d’une mesure ponctuelle de Poisson, la loi des sous-arbres qui sont attachés le long du diamètre. En application de cette décomposition des arbres de Lévy le long de leur diamètre, nous caractérisons la loi jointe de la hauteur et du diamètre des arbres de Lévy stables conditionnés à avoir une masse totale unité. Nous donnons aussi des développements asymptotiques des lois de la hauteur et du diamètre de ces arbres stables normalisés, ce qui généralise une identité due à Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) dans le cas brownien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 53, Number 2 (2017), 539-593.

Dates
Received: 10 April 2015
Revised: 20 October 2015
Accepted: 21 October 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897736

Digital Object Identifier
doi:10.1214/15-AIHP725

Zentralblatt MATH identifier
1373.60147

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60E10: Characteristic functions; other transforms 60G52: Stable processes 60G55: Point processes

Keywords
Lévy trees Height process Diameter Decomposition Asymptotic expansion Stable law

Citation

Duquesne, Thomas; Wang, Minmin. Decomposition of Lévy trees along their diameter. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 539--593. doi:10.1214/15-AIHP725. https://projecteuclid.org/euclid.aihp/1491897736


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