On a denseness result for quasi-infinitely divisible distributions

TL;DR

This paper investigates the density properties of quasi-infinitely divisible distributions in higher dimensions, establishing that unlike in one dimension, they are not dense in the space of all distributions for dimensions two and above.

Contribution

It extends the understanding of quasi-infinitely divisible distributions by proving their non-density in higher dimensions, contrasting previous results in one dimension.

Findings

Quasi-infinitely divisible distributions are dense in one dimension.

They are not dense in higher dimensions (d ≥ 2).

The result clarifies the limitations of quasi-infinite divisibility in multivariate settings.

Abstract

A probability distribution $\mu$ on $\mathbb{R}^d$ is quasi-infinitely divisible if its characteristic function has the representation $\widehat{\mu} = \widehat{\mu_1}/\widehat{\mu_2}$ with infinitely divisible distributions $\mu_1$ and $\mu_2$. In \cite[Thm. 4.1]{lindner2018} it was shown that the class of quasi-infinitely divisible distributions on $\mathbb{R}$ is dense in the class of distributions on $\mathbb{R}$ with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on $\mathbb{R}^d$ is not dense in the class of distributions on $\mathbb{R}^d$ with respect to weak convergence if $d \geq 2$.