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August 2019 Gromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson trees
Hui He, Matthias Winkel
Bernoulli 25(3): 2301-2329 (August 2019). DOI: 10.3150/18-BEJ1055

Abstract

In this paper, we study the vertex cut-trees of Galton–Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut’s vertex cut-tree of Galton–Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov–Hausdorff–Prokhorov convergence in the finite variance case of the Galton–Watson tree and its vertex cut-tree to Bertoin and Miermont’s joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut’s and Bertoin and Miermont’s Gromov–Prokhorov convergence to Gromov–Hausdorff–Prokhorov remains open for their models conditioned to have $n$ vertices.

Citation

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Hui He. Matthias Winkel. "Gromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson trees." Bernoulli 25 (3) 2301 - 2329, August 2019. https://doi.org/10.3150/18-BEJ1055

Information

Received: 1 October 2017; Revised: 1 June 2018; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066258
MathSciNet: MR3961249
Digital Object Identifier: 10.3150/18-BEJ1055

Keywords: $\mathbb{R}$-tree , Continuum random tree , Cut-tree , fragmentation at nodes , Galton–Watson tree , Gromov–Hausdorff–Prokhorov topology , invariance principle , stable tree

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 3 • August 2019
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