Single-Block Recursive Poisson–Dirichlet Fragmentations of Normalized Generalized Gamma Processes

Abstract

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for 0<α<1, and θ>−α, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters (α,1−α) to a mass partition having a Poisson–Dirichlet distribution with parameters (α,θ) leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters (α,θ+r) for r=0,1,2,⋯. Furthermore, these generate a Markovian sequence of α-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer n, which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where n=0,1.