Spinal partitions and invariance under re-rooting of continuum random trees
Bénédicte Haas, Jim Pitman, and Matthias Winkel
Source: Ann. Probab. Volume 37, Number 4 (2009), 1381-1411.
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.
Primary Subjects: 60J80
Keywords: Markov branching model; discrete tree; Poisson–Dirichlet distribution; fragmentation process; continuum random tree; spinal decomposition; random re-rooting
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1248182141
Digital Object Identifier: doi:10.1214/08-AOP434
Zentralblatt MATH identifier:
05597807
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