The Annals of Probability

Spinal partitions and invariance under re-rooting of continuum random trees

Bénédicte Haas, Jim Pitman, and Matthias Winkel

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Abstract

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

Article information

Source
Ann. Probab. Volume 37, Number 4 (2009), 1381-1411.

Dates
First available in Project Euclid: 21 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1248182141

Digital Object Identifier
doi:10.1214/08-AOP434

Mathematical Reviews number (MathSciNet)
MR2546748

Zentralblatt MATH identifier
1181.60128

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Markov branching model discrete tree Poisson–Dirichlet distribution fragmentation process continuum random tree spinal decomposition random re-rooting

Citation

Haas, Bénédicte; Pitman, Jim; Winkel, Matthias. Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. 37 (2009), no. 4, 1381--1411. doi:10.1214/08-AOP434. https://projecteuclid.org/euclid.aop/1248182141


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