DECOMPOSITION OF LEVY TREES ALONG THEIR DIAMETER
| Autor: | Thomas Duquesne, Minmin Wang |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité) |
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Statistics and Probability
Poisson distribution 01 natural sciences Measure (mathematics) Combinatorics 010104 statistics & probability symbols.namesake Mathematics::Probability 60E07 Random tree FOS: Mathematics Continuum (set theory) 0101 mathematics diameter stable law Brownian motion Mathematics 60J80 asymptotic expansion decomposition Probability (math.PR) 010102 general mathematics [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Metric space height process symbols Tree (set theory) 60G55 Statistics Probability and Uncertainty Asymptotic expansion 60E10 Mathematics - Probability 60G52 Lévy trees |
| Zdroj: | Ann. Inst. H. Poincaré Probab. Statist. 53, no. 2 (2017), 539-593 Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2017, ⟨10.1214/15-AIHP725⟩ |
| ISSN: | 0246-0203 1778-7017 |
| DOI: | 10.1214/15-AIHP725⟩ |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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