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September 2018 Recursive construction of continuum random trees
Franz Rembart, Matthias Winkel
Ann. Probab. 46(5): 2715-2748 (September 2018). DOI: 10.1214/17-AOP1237

Abstract

We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related structures. We prove the existence of these CRTs as a new application of the fixpoint method for recursive distribution equations formalised in high generality by Aldous and Bandyopadhyay.

We apply this recursive method to show the convergence to CRTs of various tree growth processes. We note alternative constructions of existing self-similar CRTs in the sense of Haas, Miermont and Stephenson, and we give for the first time constructions of random compact $\mathbb{R}$-trees that describe the genealogies of Bertoin’s self-similar growth fragmentations. In forthcoming work, we develop further applications to embedding problems for CRTs, providing a binary embedding of the stable line-breaking construction that solves an open problem of Goldschmidt and Haas.

Citation

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Franz Rembart. Matthias Winkel. "Recursive construction of continuum random trees." Ann. Probab. 46 (5) 2715 - 2748, September 2018. https://doi.org/10.1214/17-AOP1237

Information

Received: 1 July 2016; Revised: 1 October 2017; Published: September 2018
First available in Project Euclid: 24 August 2018

zbMATH: 06964347
MathSciNet: MR3846837
Digital Object Identifier: 10.1214/17-AOP1237

Subjects:
Primary: 60J05, 60J80

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • September 2018
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