Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Convergence of bi-measure $\mathbb{R}$-trees and the pruning process

Wolfgang Löhr, Guillaume Voisin, and Anita Winter

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Abstract

In (Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637–686) a tree-valued Markov chain is derived by pruning off more and more subtrees along the edges of a Galton–Watson tree. More recently, in (Ann. Probab. 40 (2012) 1167–1211), a continuous analogue of the tree-valued pruning dynamics is constructed along Lévy trees. In the present paper, we provide a new topology which allows to link the discrete and the continuous dynamics by considering them as instances of the same strong Markov process with different initial conditions. We construct this pruning process on the space of so-called bi-measure trees, which are metric measure spaces with an additional pruning measure. The pruning measure is assumed to be finite on finite trees, but not necessarily locally finite. We also characterize the pruning process analytically via its Markovian generator and show that it is continuous in the initial bi-measure tree. A series of examples is given, which include the finite variance offspring case where the pruning measure is the length measure on the underlying tree.

Résumé

Dans (Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637–686), les auteurs obtiennent une chaîne de Markov à valeurs arbres en élaguant de plus en plus de sous-arbres le long des nœuds d’un arbre de Galton–Watson. Plus récemment dans (Ann. Probab. 40 (2012) 1167–1211), un analogue continu de la dynamique d’élagage à valeurs arbres est construit sur des arbres de Lévy. Dans cet article, nous présentons une nouvelle topologie qui permet de relier les dynamiques discrètes et continues en les considérant comme des exemples du même processus de Markov fort avec des conditions initiales différentes. Nous construisons ce processus d’élagage sur l’espace des arbres appelés bi-mesurés, qui sont des espaces métriques mesurés avec une mesure d’élagage additionnelle. La mesure d’élagage est supposée finie sur les arbres finis, mais pas nécessairement localement finie. De plus, nous caractérisons analytiquement le processus d’élagage par son générateur infinitésimal et montrons qu’il est continu en son arbre bi-mesuré initial. Plusieurs exemples sont donnés, notamment le cas d’une loi de reproduction à variance finie où la mesure d’élagage est la mesure des longueurs sur l’arbre sous-jacent.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1342-1368.

Dates
Received: 19 April 2013
Revised: 21 January 2014
Accepted: 3 June 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432045

Digital Object Identifier
doi:10.1214/14-AIHP628

Mathematical Reviews number (MathSciNet)
MR3414450

Zentralblatt MATH identifier
1339.60123

Subjects
Primary: 60F05: Central limit and other weak theorems 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60J25: Continuous-time Markov processes on general state spaces 05C05: Trees 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 60G55: Point processes

Keywords
Tree-valued Markov process CRT Real trees Pruning procedure Pointed Gromov-weak topology Prohorov metric Non-locally finite measures

Citation

Löhr, Wolfgang; Voisin, Guillaume; Winter, Anita. Convergence of bi-measure $\mathbb{R}$-trees and the pruning process. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1342--1368. doi:10.1214/14-AIHP628. https://projecteuclid.org/euclid.aihp/1445432045


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