Pruning Galton-Watson Trees and Tree-valued Markov Processes

Romain Abraham; Jean-François Delmas; Hui He

hal-00497035, version 2

Pruning Galton-Watson Trees and Tree-valued Markov Processes

Romain Abraham () ¹, Jean-François Delmas (, ) ², Hui He ^1, 3

Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques 48, 3 (2012) 688-705

Abstract: We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process $\{ {\cal G}(u)\}$ by pruning Galton-Watson trees and an analogous process $\{{\cal G}^*(u)\}$ by pruning a critical or subcritical Galton-Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process $\{{\cal G}(u)\}$ run until its ascension time has a representation in terms of $\{{\cal G}^*(u)\}$. A similar result was obtained by Aldous and Pitman (1998) in the special case of Poisson offspring distributions where they considered uniform pruning of Galton-Watson trees by adding marks on the edges of trees.

1: Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO)
Université d'Orléans – CNRS : UMR7349
2: Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
Ecole des Ponts ParisTech
3: School of Mathematical Sciences
Beijing Normal University / Beijing

hal-00497035, version 2
http://hal.archives-ouvertes.fr/hal-00497035
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From: Romain Abraham <>
Submitted on: Monday, 7 February 2011 11:53:18
Updated on: Wednesday, 27 June 2012 14:52:37

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arXiv: 1007.0370

DOI: 10.1214/11-AIHP423

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