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 Lindner et al. (2018, Thm. 4.1) 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 $d\ge 2$.

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关于拟无限可整分布的密度结果

概率分布 $\mu$ 上 ${\mathbb{[R}}^d$ 如果其特征函数具有表示形式，则是准无限可整的 $\widehat{\mu }=\widehat{{\mu }_{\mathrm{1个}}}∕\widehat{{\mu }_{\mathrm{2个}}}$ 具有无限可整的分布 ${\mu }_{\mathrm{1个}}$ 和 ${\mu }_{\mathrm{2个}}$。在林德纳（Lindner）等人中。（2018，Thm.4.1）研究表明，$\mathbb{[R}$ 在分布的类别上是密集的 $\mathbb{[R}$关于弱收敛。在本文中，我们证明了上的拟无限可分分布的类${\mathbb{[R}}^d$ 在分布类别上并不密集 ${\mathbb{[R}}^d$ 关于弱收敛，如果 $d\ge \mathrm{2个}$。