On a denseness result for quasi-infinitely divisible distributions

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 $d geq 2$.