Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 512-532.

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.

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.

DOI : 10.1214/13-AIHP594
Classification : 60J80
Mots-clés : Markov branching trees, Galton–Watson trees, continuum random tree 
@article{AIHPB_2015__51_2_512_0,
     author = {Rizzolo, Douglas},
     title = {Scaling limits of {Markov} branching trees and {Galton{\textendash}Watson} trees conditioned on the number of vertices with out-degree in a given set},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {512--532},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     doi = {10.1214/13-AIHP594},
     mrnumber = {3335013},
     zbl = {1319.60170},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/13-AIHP594/}
}
TY  - JOUR
AU  - Rizzolo, Douglas
TI  - Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 512
EP  - 532
VL  - 51
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/13-AIHP594/
DO  - 10.1214/13-AIHP594
LA  - en
ID  - AIHPB_2015__51_2_512_0
ER  - 
%0 Journal Article
%A Rizzolo, Douglas
%T Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 512-532
%V 51
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/13-AIHP594/
%R 10.1214/13-AIHP594
%G en
%F AIHPB_2015__51_2_512_0
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. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 512-532. doi : 10.1214/13-AIHP594. http://www.numdam.org/articles/10.1214/13-AIHP594/

[1] R. Abraham and J.-F. Delmas. Local limits of conditioned Galton–Watson trees I: The infinite spine case. Electron. J. Probab. 19 (2) (2014) 1–19 (electronic). DOI:10.1214/EJP.v19-2747 | MR | Zbl

[2] D. Aldous. The continuum random tree. III. Ann. Probab. 21 (1) (1993) 248–289. | MR | Zbl

[3] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. | DOI | MR | Zbl

[4] B. Haas and G. Miermont. The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (4) (2004) 57–97. | MR | Zbl

[5] B. Haas and G. Miermont. Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 (6) (2012) 2589–2666. | MR | Zbl

[6] B. Haas, G. Miermont, J. Pitman and M. Winkel. Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 (5) (2008) 1790–1837. | MR | Zbl

[7] I. Kortchemski. Invariance principles for Galton–Watson trees conditioned on the number of leaves. Stochastic Process. Appl. 122 (9) (2012) 3126–3172. | MR | Zbl

[8] J.-F. Le Gall. Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 (1) (2006) 35–62. | Numdam | MR | Zbl

[9] J.-F. Le Gall. Itô’s excursion theory and random trees. Stochastic Process. Appl. 120 (5) (2010) 721–749. | MR | Zbl

[10] N. Minami. On the number of vertices with a given degree in a Galton–Watson tree. Adv. in Appl. Probab. 37 (1) (2005) 229–264. | MR | Zbl

[11] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Springer, Berlin, 2006. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. | MR | Zbl

[12] D. Rizzolo. Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set, 2011. Available at arXiv:1105.2528v1. | Numdam | MR

Cité par Sources :