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Local limits of Galton–Watson trees conditioned on the number of protected nodes

  • Romain Abraham (a1), Aymen Bouaziz (a2) and Jean-François Delmas (a3)

Abstract

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes.

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Corresponding author

* Postal address: Laboratoire MAPMO, Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France. Email address: romain.abraham@univ-orleans.fr
** Postal address: Institut préparatoire aux études scientifiques et techniques, Université de Tunis El Manar, 2070 La Marsa, Tunis, Tunisie.
*** Postal address: CERMICS, Ecole des Ponts, Université Paris-Est, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France.

References

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[1] Abraham, R. and Delmas, J.-F. (2014).Local limits of conditioned Galton–Watson trees: the infinite spine case.Electron. J. Prob. 19, 19 pp.
[2] Abraham, R. and Delmas, J.-F. (2015).An introduction to Galton–Watson trees and their local limits. Lecture given in Hammamet, December 2014. Available at https://arxiv.org/abs/1506.05571.
[3] Devroye, L. and Janson, S. (2014).Protected nodes and fringe subtrees in some random trees.Electron. Commun. Prob. 19, 10 pp.
[4] He, X. (2015).Local convergence of critical random trees and continuous-state branching processes. Preprint. Available at https://arxiv.org/abs/1503.00951.
[5] Janson, S. (2016).Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees.Random Structures Algorithms 48,57101.
[6] Kesten, H. (1986).Subdiffusive behavior of random walk on a random cluster.Ann. Inst. H. Poincaré Prob. Statist. 22,425487.
[7] Neveu, J. (1986).Arbres et processus de Galton–Watson..Ann. Inst. H. Poincaré Prob. Statist. 22,199207.
[8] Rizzolo, D. (2015).Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set.Ann. Inst. H. Poincaré Prob. Statist. 51,512532.

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Local limits of Galton–Watson trees conditioned on the number of protected nodes

  • Romain Abraham (a1), Aymen Bouaziz (a2) and Jean-François Delmas (a3)

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