# On quasi-infinitely divisible random measures

## TL;DR

This paper investigates quasi-infinitely divisible (QID) random measures, showing their density in the space of all completely random measures (CRMs), establishing a Lévy-Khintchine formulation, and exploring implications for Bayesian nonparametrics.

## Contribution

It proves the density of QID CRMs in the space of all CRMs and establishes a Lévy-Khintchine representation with a one-to-one law correspondence.

## Key findings

- QID CRMs are dense in all CRMs with respect to distribution convergence.
- QID CRMs possess a Lévy-Khintchine type representation.
- Results have implications for Bayesian nonparametric models.

## Abstract

Quasi-infinitely divisible (QID) distributions have been recently introduced by Lindner, Pan and Sato (\textit{Trans.~Amer.~Math.~Soc.}~\textbf{370}, 8483-8520 (2018)). A random variable $X$ is QID if and only if there exist two infinitely divisible (ID) random variables $Y$ and $Z$ s.t.~$X+Y\stackrel{d}{=}Z$ and $Y$ is independent of $X$. In this work, we show that a family of QID completely random measures (CRMs) is dense in the space of all CRMs with respect to convergence in distribution. We further demonstrate that the elements of this family posses a L\'{e}vy-Khintchine formulation and that there exists a one to one correspondence between their law and certain characteristic pairs. We prove the same results also for the class of point processes with independent increments. In the second part of the paper, we show the relevance of these results in the general Bayesian nonparametric framework based on CRMs developed by Broderick, Wilson and Jordan (\textit{Bernoulli}, \textbf{24}, 3181-3221 (2018)).