Gibbs fragmentation trees

Abstract

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $\beta >-2$ with respect to the $\mathrm{beta}(\beta +1,\beta +1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\mathbb{N}$, with an extended parameter range $0\le \alpha \le 1$, $\theta \ge -2\alpha$ and $\alpha <0$, $\theta =-m\alpha$, $m\in \mathbb{N}$.

Published in at http://dx.doi.org/10.3150/08-BEJ134 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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