On a denseness result for quasi-infinitely divisible distributions

A probability distribution $μ$ on $\mathbb{R}^d$ is quasi-infinitely divisible if its characteristic function has the representation $\widehatμ = \widehat{μ_1}/\widehat{μ_2}$ with infinitely divisible distributions $μ_1$ and $μ_2$. 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 $d \geq 2$.

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