Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Invariance principles for spatial multitype Galton–Watson trees

Grégory Miermont

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Abstract

We prove that critical multitype Galton–Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the Brownian snake, under some moment assumptions.

Résumé

Nous montrons que les arbres de Galton–Watson multitypes, dont les lois de reproduction sont irrductibles et de matrices de covariance finies, admettent pour limite d’chelle l’arbre continu brownien. La clef de notre tude est une dcomposition ancestrale pour les arbres multitypes marqus, et une mthode par rcurrence sur le nombre de types. Nous couplons ensuite la structure gnalogique avec des dplacements spaciaux, dont la loi de saut peut dpendre localement de la structure de l’arbre, et nous montrons que les arbres spatiaux obtenus convergent vers le serpent brownien, sous certaines hypothses de moments.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 6 (2008), 1128-1161.

Dates
First available in Project Euclid: 21 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1227287568

Digital Object Identifier
doi:10.1214/07-AIHP157

Mathematical Reviews number (MathSciNet)
MR2469338

Zentralblatt MATH identifier
1178.60058

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles

Keywords
Multitype Galton–Watson tree Discrete snake Invariance principle Brownian tree Brownian snake

Citation

Miermont, Grégory. Invariance principles for spatial multitype Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 6, 1128--1161. doi:10.1214/07-AIHP157. https://projecteuclid.org/euclid.aihp/1227287568


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