Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set

Douglas Rizzolo

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Abstract

We obtain scaling limits for Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. This extends recent results of Haas and Miermont in (Ann. Probab. 40 (2012) 2589–2666), which considered the case when the size of a tree is either its number of leaves or its number of vertices. We use our result to prove that the scaling limit of finite variance Galton–Watson trees conditioned on the number of nodes whose out-degree lies in a given set is the Brownian continuum random tree. The key to applying our result for Markov branching trees to conditioned Galton–Watson trees is a generalization of the classical Otter–Dwass formula. This is obtained by showing that the number of vertices in a Galton–Watson tree whose out-degree lies in a given set is distributed like the number of vertices in a Galton–Watson tree with a related offspring distribution.

Résumé

Nous obtenons des limites d’échelle pour des arbres branchants markoviens dont la taille est égale au nombre de noeuds dont le degré sortant appartient à un ensemble fixé. Ceci étend des résultats récents de Haas et Miermont dans (Ann. Probab. 40 (2012) 2589–2666), qui ont considéré le cas où la taille d’un arbre est le nombre de ses feuilles ou le nombre des sommets. Nous utilisons nos résultats pour prouver que la limite d’échelle d’arbres de Galton–Watson conditionnés par le nombre de noeuds dont le degré sortant appartient à un ensemble donné est l’arbre brownien continu. La clé pour appliquer notre résultat pour les arbres branchants markoviens à des arbres de Galton–Watson conditionnés est une généralisation de la formule classique de Otter–Dwass. Ceci est obtenu en montrant que le nombre de sommets d’un arbre de Galton–Watson dont le degré sortant appartient à un ensemble donné est distribué comme le nombre de sommets dans un arbre de Galton–Watson avec une loi de reproduction appropriée.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 2 (2015), 512-532.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672679

Digital Object Identifier
doi:10.1214/13-AIHP594

Mathematical Reviews number (MathSciNet)
MR3335013

Zentralblatt MATH identifier
1319.60170

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Markov branching trees Galton–Watson trees Continuum random tree

Citation

Rizzolo, Douglas. 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 Probab. Statist. 51 (2015), no. 2, 512--532. doi:10.1214/13-AIHP594. https://projecteuclid.org/euclid.aihp/1428672679


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References

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