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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: 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.
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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 

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References

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