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September 2009 Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions
Jim Pitman, Matthias Winkel
Ann. Probab. 37(5): 1999-2041 (September 2009). DOI: 10.1214/08-AOP445


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.


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Jim Pitman. Matthias Winkel. "Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions." Ann. Probab. 37 (5) 1999 - 2041, September 2009.


Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1189.60162
MathSciNet: MR2561439
Digital Object Identifier: 10.1214/08-AOP445

Primary: 60J80

Keywords: Chinese restaurant process , Continuum random tree , Markov branching model , phylogenetic tree , Poisson–Dirichlet composition , recursive random tree , regenerative composition , ℝ-tree , Self-similar fragmentation

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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