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

Lancelot James

doi:10.3390/math10040561

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.

Highlights

Let Z = ( Zr, r ≥ 0) denote a Markov chain characterized by a stationary transition density Zr | Zr−1 = z given for y > z and 0 < α < 1: Academic Editors: Emanuele Dolera and Federico Bassetti
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We shall primarily focus on the case of n = 0, 1, corresponding to the generalized gamma density and its sized biased distribution, which yields the most explicit results

Summary

Introduction

Let ( P ) := (( P ), ` ≥ 1) ∼ PD(α, θ ) correspond in distribution to the ranked lengths of excursion of a generalized Bessel bridge on [0, 1], as described and defined in [1,4]. We obtain results for the case where (( P,r ), Zr ; r ≥ 0) is such that ( P,0 )∼Pα (λ), which is when ( P,0 ) corresponds to the ranked normallized jumps of a generalized gamma process, (τα (y); y ≥ 0), and its size-biased generalizations. We shall primarily focus on the case of n = 0, 1, corresponding to the generalized gamma density and its sized biased distribution, which yields the most explicit results. Let (U ) denote a sequence of iid Uniform[0, 1] variables independent of ( P ) ∼ PD(α, θ ); the random distribution. See [1,4,24] and references therein for various interpretations of (4)

DGM Fragmentation

Remarks

Results

Proofs of Propositions 2 and 3