A limit theorem for the contour process of condidtioned Galton--Watson trees
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Thomas Duquesne [1]
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[1] École Normale Supérieure de Cachan
École Normale Supérieure de Cachan
Arrondissement de L'Haÿ-les-Roses, Francia
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 2, 2003, págs. 996-1027
- Idioma: inglés
- DOI: 10.1214/aop/1048516543
- Texto completo no disponible (Saber más ...)
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Resumen
In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index α∈(1,2]. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one ancestor and conditioned on the total progeny, converges in a functional sense, to a new process: the normalized excursion of the continuous height process associated with the α-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case α=2, the limiting process is the normalized Brownian excursion that codes the continuum random tree: the result is due to Aldous who used a different method.
