Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions
Jim Pitman and Matthias Winkel
Source: Ann. Probab. Volume 37, Number 5 (2009), 1999-2041.
Abstract
We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions.
Primary Subjects: 60J80
Keywords: Regenerative composition; Poisson–Dirichlet composition; Chinese Restaurant Process; Markov branching model; self-similar fragmentation; continuum random tree; ℝ-tree; recursive random tree; phylogenetic tree
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1253539862
Digital Object Identifier: doi:10.1214/08-AOP445
Mathematical Reviews number (MathSciNet):
MR2561439
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