DECOMPOSITION OF LEVY TREES ALONG THEIR DIAMETER

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 and Le Jan (1998) 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 ∈ (0, ∞) 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 (1983) in the Brownian case.
Type de document :
Pré-publication, Document de travail
2015
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https://hal.archives-ouvertes.fr/hal-01132231
Contributeur : Thomas Duquesne <>
Soumis le : lundi 16 mars 2015 - 19:28:50
Dernière modification le : jeudi 27 avril 2017 - 09:46:04
Document(s) archivé(s) le : mercredi 17 juin 2015 - 15:35:34

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  • HAL Id : hal-01132231, version 1
  • ARXIV : 1503.05069

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INSMI | UPMC | USPC | PMA

Citation

Thomas Duquesne, Minmin Wang. DECOMPOSITION OF LEVY TREES ALONG THEIR DIAMETER. 2015. <hal-01132231>

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