On decomposition problem for distribution functions of class $\boldsymbol{Q}$

TL;DR

This paper investigates the decomposition properties of a new class of distribution functions called nd their relation to infinitely divisible distributions, providing a general answer to an open question about their factorization.

Contribution

It proves that the decomposition property holds for the class nd extends the results to the class nd infinitely divisible distributions without additional assumptions.

Findings

The decomposition property holds for the class nd general settings.

The results apply to both nd infinitely divisible distributions.

Answers an open question posed in 2018 about distribution function factorization.

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--Khinchine type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$, being a natural extension of the class $\boldsymbol{I}$ of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan and Sato formulated the open question: is it true that if $F\in\boldsymbol{Q}$ and $F=F_1*F_2$ with some distribution functions $F_1$ and $F_2$, then $F_1\in\boldsymbol{Q}$ and $F_2\in\boldsymbol{Q}$? There are some positive results under special assumptions on the type of $F$. In this paper, we answer the question in a general setting without any additional assumptions. We also consider the same question but with the stronger assumption that $F\in\boldsymbol{I}$.