On Quasi-Infinitely Divisible Distributions with a Point Mass

An infinitely divisible distribution on $\mathbb{R}$ is a probability measure $\mu$ such that the characteristic function $\hat{\mu}$ has a L\'{e}vy-Khintchine representation with characteristic triplet $(a,\gamma, \nu)$, where $\nu$ is a L\'{e}vy measure, $\gamma\in\mathbb{R}$ and $a\ge 0$. A natural extension of such distributions are quasi-infinitely distributions. Instead of a L\'{e}vy measure, we assume that $\nu$ is a "signed L\'{e}vy measure", for further information on the definition see [\ref{Lindner}]. We show that a distribution $\mu=p\delta_{x_0}+(1-p)\mu_{ac}$ with $p>0$ and $x_0 \in \mathbb{R}$, where $\mu_{ac}$ is the absolutely continuous part, is quasi-infinitely divisible if and only if $\hat{\mu}(z)\neq0$ for every $z\in\mathbb{R}$. We apply this to show that certain variance mixtures of mean zero normal distributions are quasi-infinitely divisible distributions, and we give an example of a quasi-infinitely divisible distribution that is not continuous but has infinite quasi-L\'{e}vy measure. Furthermore, it is shown that replacing the signed L\'{e}vy measure by a seemingly more general complex L\'{e}vy measure does not lead to new distributions. Last but not least it is proven that the class of quasi-infinitely divisible distributions is not open, but path-connected in the space of probability measures with the Prokhorov metric.