• Bernoulli
  • Volume 14, Number 4 (2008), 988-1002.

Gibbs fragmentation trees

Peter McCullagh, Jim Pitman, and Matthias Winkel

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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 $β>−2$ with respect to the beta $(β+1, β+1)$ 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 $ℕ$, with an extended parameter range $0≤α≤1$, $θ≥−2α$ and $α<0$, $θ=−mα$, $m∈ℕ$.

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Bernoulli, Volume 14, Number 4 (2008), 988-1002.

First available in Project Euclid: 6 November 2008

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Aldous’ beta-splitting model Gibbs distribution Markov branching model Poisson–Dirichlet distribution


McCullagh, Peter; Pitman, Jim; Winkel, Matthias. Gibbs fragmentation trees. Bernoulli 14 (2008), no. 4, 988--1002. doi:10.3150/08-BEJ134.

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