Annals of Applied Probability

The vertex-cut-tree of Galton–Watson trees converging to a stable tree

Daphné Dieuleveut

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Abstract

We consider a fragmentation of discrete trees where the internal vertices are deleted independently at a rate proportional to their degree. Informally, the associated cut-tree represents the genealogy of the nested connected components created by this process. We essentially work in the setting of Galton–Watson trees with offspring distribution belonging to the domain of attraction of a stable law of index $\alpha\in(1,2)$. Our main result is that, for a sequence of such trees $\mathcal{T}_{n}$ conditioned to have size $n$, the corresponding rescaled cut-trees converge in distribution to the stable tree of index $\alpha$, in the sense induced by the Gromov–Prokhorov topology. This gives an analogue of a result obtained by Bertoin and Miermont in the case of Galton–Watson trees with finite variance.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 4 (2015), 2215-2262.

Dates
Received: December 2013
Revised: May 2014
First available in Project Euclid: 21 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1432212441

Digital Object Identifier
doi:10.1214/14-AAP1047

Mathematical Reviews number (MathSciNet)
MR3349006

Zentralblatt MATH identifier
1319.60167

Subjects
Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Galton–Watson tree cut-tree stable continuous random tree

Citation

Dieuleveut, Daphné. The vertex-cut-tree of Galton–Watson trees converging to a stable tree. Ann. Appl. Probab. 25 (2015), no. 4, 2215--2262. doi:10.1214/14-AAP1047. https://projecteuclid.org/euclid.aoap/1432212441


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