Growth of Lévy trees |
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Authors: | Thomas Duquesne Matthias Winkel |
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Institution: | (1) Département de Mathématiques, Université Paris 11, 91405 Orsay Cedex, France;(2) Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK |
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Abstract: | We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state
branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch
lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family
with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees
and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state
branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of
a Poisson sampling directed by the mass measure.
T. Duquesne is supported by NSF Grants DMS-0203066 and DMS-0405779. M. Winkel is supported by Aon and the Institute of Actuaries,
EPSRC Grant GR/T26368/01, le département de mathématique de l’Université d’Orsay and NSF Grant DMS-0405779. |
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Keywords: | Tree-valued Markov process Galton– Watson branching process Genealogy Continuous-state branching process Percolation Gromov– Hausdorff topology Continuum random tree Edge lengths Real tree |
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