Gibbs fragmentation trees

Details Gibbs fragmentation trees Gibbs fragmentation trees

2007-04-06

arxiv.org/abs/0704.0945v2

Bernoulli 2008, Vol. 14, No. 4, 988-1002

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/08-BEJ134 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ134

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 ${\rm beta}(\beta+1,\beta+1)$ 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 $0\le\alpha\le1$, $\theta\ge-2\alpha$ and $\alpha<0$, $\theta =-m\alpha$, $m\in \mathbb {N}$.

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