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

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for 0 < alpha < 1, and theta > -alpha, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson-Dirichlet distribution with parameters (alpha,1-alpha) to a mass partition having a Poisson-Dirichlet distribution with parameters (alpha, theta) leads to a remarkable nested family of Poisson-Dirichlet distributed mass partitions with parameters (alpha,theta+r) for r = 0, 1, 2, ... . Furthermore, these generate a Markovian sequence of alpha-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 Polya 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.