The Annals of Probability

Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions

Jim Pitman and Matthias Winkel

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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.

Article information

Source
Ann. Probab. Volume 37, Number 5 (2009), 1999-2041.

Dates
First available in Project Euclid: 21 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1253539862

Digital Object Identifier
doi:10.1214/08-AOP445

Mathematical Reviews number (MathSciNet)
MR2561439

Zentralblatt MATH identifier
1189.60162

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

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

Citation

Pitman, Jim; Winkel, Matthias. Regenerative tree growth: Binary self-similar continuum random trees and Poisson–Dirichlet compositions. Ann. Probab. 37 (2009), no. 5, 1999--2041. doi:10.1214/08-AOP445. https://projecteuclid.org/euclid.aop/1253539862


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