The class $\boldsymbol{Q}$ and mixture distributions with dominated continuous singular parts

TL;DR

This paper introduces the class Q of distribution functions with rational-infinite divisibility, providing criteria for membership when a continuous singular part is dominated by the discrete part, and explores their Lévy-type representations.

Contribution

It extends the class of infinitely divisible distributions to include those with dominated continuous singular parts and provides a new criterion for their characterization.

Findings

Established a criterion for class Q membership with dominated continuous singular parts.

Described the Lévy-type characteristic triplet for these distributions.

Solved the decomposition problem posed by Lindner, Pan, and Sato within this class.

Abstract

We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the L\'evy-type representation with a ``signed spectral measure''. This class is a wide natural extension of the fundamental class of infinitely divisible distribution functions and it is actively studied now. We are interested in conditions for a distribution function $F$ to belong to the class $\boldsymbol{Q}$ for the unexplored case, where $F$ may have a continuous singular part. We propose a criterion under the assumption that the continuous singular part of $F$ is dominated by the discrete part in a certain sense. The criterion generalizes the previous results by Alexeev and Khartov for discrete probability laws and the results by Berger and Kutlu for the mixtures of discrete and absolutely continuous laws. In addition, we describe the characteristic triplet of the corresponding L\'evy-type representation, which may contain some continuous singular part. We also show that the assumption of the dominated continuous singular part cannot be simply omitted or even slightly extended (without some special assumptions). We apply the general criterion to some interesting particular examples. We also positively solve the decomposition problem stated by Lindner, Pan and Sato within the considered case.