On a denseness result for quasi-infinitely divisible distributions

Abstract

A probability distribution $\mu$ on ${\mathbb{R}}^d$ is quasi-infinitely divisible if its characteristic function has the representation $\hat{\mu }=\widehat{{\mu }_1}/\widehat{{\mu }_2}$ 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 $d\ge 2$.