Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg