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 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 œ ℕ. © 2008 ISI/BS.