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

Abstract

We consider a new class $\mathbf{Q}$ 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 $F_1=F\ast F_2$. A distribution function of the class $\mathbf{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy--Khinchine type representation with a ``signed spectral measure''. The class $\mathbf{Q}$, being a natural extension of the class $\mathbf{I}$ 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 $F\in \mathbf{Q}$ and $F=F_1\ast F_2$ with some distribution functions $F_1$ and $F_2$, then $F_1\in \mathbf{Q}$ and $F_2\in \mathbf{Q}$? 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 $F\in \mathbf{I}$.