On a Gibbs characterization of normalized generalized Gamma processes

TL;DR

This paper offers an alternative derivation of a Gibbs characterization for normalized generalized Gamma processes using exponential tilting of Poisson-Kingman models, and explores conditions for exchangeable Gibbs partitions.

Contribution

It provides a new derivation method for the Gibbs characterization and investigates the existence of normalized random measures for specific Gibbs partitions.

Findings

Alternative derivation of Gibbs characterization using Poisson-Kingman models

Conditions for existence of normalized random measures for Gibbs partitions

Extension of previous results on exchangeable Gibbs partitions

Abstract

We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Pr\"unster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted Poisson-Kingman models stated in Pitman (2003). We also provide a completion of this result investigating the existence of normalized random measures inducing exchangeable Gibbs partitions of type $\alpha \in (-\infty, 0]$.