Alexey Khartov | Searcharxiv

arXiv2025

Denseness in total variation and the class of rational-infinitely divisible distributions

We study a new class of so-called rational-infinitely (or quasi-infinitely) divisible probability laws on the real line. The characteristic functions of these distributions are ratios of the characteristic functions of classical infinitely divisible laws and they admit Lévy--Khinchine type representations with ``signed spectral measures''. This class is rather wide and it has a lot of nice properties. For instance, this class is dense in the family of all (univariate) probability laws with respect to weak convergence. In this paper, we consider the questions concerning a denseness of this class with respect to convergence in total variation. The problem is considered separately for different types of probability laws taking into account the supports of the distributions. A series of ``positive'' and ``negative'' results are obtained.

math.PR↗

arXiv2025

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

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.

math.PR↗