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

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 .