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 $α<0$, $θ=−mα$, $m∈ℕ$.