On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts

Abstract

We consider the new class $\boldsymbol{Q}$ of rational-infinitely (or quasi-infinitely) divisible distribution functions on the real line. By definition, $F\in \boldsymbol{Q}$ if there are some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$, where ``$*$'' is the convolution. The characteristic function of such $F$ admits the Lévy--Khintchine-type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$ is a significant extension of the family of infinitely divisible distribution functions and it have already found some applications in several areas. So there is an active interest in this class. In particular, a lot of results have recently appeared on the problem of belonging to the class $\boldsymbol{Q}$ in terms of characteristic functions. In the paper, we continue this series of results by proposing two necessary conditions for distribution functions from $\boldsymbol{Q}$ with non-zero discrete parts. Namely, the characteristic functions of such a distribution function and its discrete part are always separated from zero.