The Annals of Probability

Recursive construction of continuum random trees

Franz Rembart and Matthias Winkel

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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.

Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2715-2748.

Dates
Received: July 2016
Revised: October 2017
First available in Project Euclid: 24 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1535097638

Digital Object Identifier
doi:10.1214/17-AOP1237

Mathematical Reviews number (MathSciNet)
MR3846837

Zentralblatt MATH identifier
06964347

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J05: Discrete-time Markov processes on general state spaces

Keywords
String of beads $\mathbb{R}$-tree continuum random tree self-similar tree stable tree recursive distribution equation tree growth growth fragmentation Hausdorff dimension

Citation

Rembart, Franz; Winkel, Matthias. Recursive construction of continuum random trees. Ann. Probab. 46 (2018), no. 5, 2715--2748. doi:10.1214/17-AOP1237. https://projecteuclid.org/euclid.aop/1535097638


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