Exchangeable random partitions from max-infinitely-divisible distributions

TL;DR

This paper explores a class of max-infinitely-divisible distributions with exchangeable hitting partitions, providing explicit formulas and connecting them to well-known Poisson--Dirichlet partitions, thereby advancing understanding of their structure.

Contribution

It introduces a new class of max-i.d. laws with exchangeable hitting partitions derived from Lévy subordinators, linking them to Poisson--Dirichlet distributions.

Findings

Hitting partitions for specific max-i.d. laws match Poisson--Dirichlet distributions.

Explicit formulas for distributions of these partitions are derived.

Exchangeable hitting partitions are characterized for multivariate α-logistic and related families.

Abstract

The hitting partitions are random partitions that arise from the investigation of so-called hitting scenarios of max-infinitely-divisible (max-i.d.)~distributions. We study a class of max-i.d.~laws with exchangeable hitting partitions obtained by size-biased sampling from the jumps of a L\'evy subordinator. We obtain explicit formulae for the distributions of these partitions in the case of the multivariate $\alpha$-logistic and another family of exchangeable max-i.d.\ distributions. Specifically, the hitting partitions for these two cases are shown to coincide with the well-known Poisson--Dirichlet partitions ${\rm PD}(\alpha,0),\ \alpha\in (0,1)$ and ${\rm PD}(0,\theta),\ \theta>0$.