Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T05:50:51.913Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Graph theory Probability theory on algebraic and topological structures Markov processes

Published online by Cambridge University Press:  04 April 2017

Romain Abraham ,
Aymen Bouaziz  and
Jean-François Delmas
Romain Abraham*
Affiliation:
Université d’Orléans
Aymen Bouaziz*
Affiliation:
Université de Tunis El Manar
Jean-François Delmas*
Affiliation:
Ecole des Ponts
*
* Postal address: Laboratoire MAPMO, Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France. Email address: [email protected]
** 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.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Galton--Watson treerandom treelocal limitprotected node

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60B10: Convergence of probability measures
Secondary: 05C05: Trees
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 55 - 65
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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.Google Scholar
[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.Google Scholar
[3] Devroye, L. and Janson, S. (2014).Protected nodes and fringe subtrees in some random trees.Electron. Commun. Prob. 19, 10 pp.CrossRefGoogle Scholar
[4] He, X. (2015).Local convergence of critical random trees and continuous-state branching processes. Preprint. Available at https://arxiv.org/abs/1503.00951.Google Scholar
[5] Janson, S. (2016).Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees.Random Structures Algorithms 48,57–101.CrossRefGoogle Scholar
[6] Kesten, H. (1986).Subdiffusive behavior of random walk on a random cluster.Ann. Inst. H. Poincaré Prob. Statist. 22,425–487.Google Scholar
[7] Neveu, J. (1986).Arbres et processus de Galton–Watson..Ann. Inst. H. Poincaré Prob. Statist. 22,199–207.Google Scholar
[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. 51,512–532.Google Scholar