Electronic Journal of Probability

A line-breaking construction of the stable trees

Christina Goldschmidt and Bénédicte Haas

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We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $\alpha = 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 16, 24 pp.

Accepted: 24 February 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 05C05: Trees
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G52: Stable processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

stable Lévy trees line-breaking generalized Mittag-Leffler distributions Dirichlet distributions

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Goldschmidt, Christina; Haas, Bénédicte. A line-breaking construction of the stable trees. Electron. J. Probab. 20 (2015), paper no. 16, 24 pp. doi:10.1214/EJP.v20-3690. https://projecteuclid.org/euclid.ejp/1465067122

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