Three Upsilon Transforms Related to Tempered Stable Distributions

TL;DR

This paper explores three upsilon transforms linked to tempered stable distributions, detailing their properties, domains, and how they can convert Lévy measures between different tempered stable classes.

Contribution

It introduces and characterizes three specific upsilon transforms, demonstrating their ability to relate various tempered stable distributions through measure transformations.

Findings

Characterization of the domains of the three upsilon transforms.

Representation of these transforms as compositions of each other.

Transformations of Lévy measures between different tempered stable classes.

Abstract

We discuss the properties of three upsilon transforms, which are related to the class of $p$-tempered $\alpha$-stable ($TS^p_\alpha$) distributions. In particular, we characterize their domains and show how they can be represented as compositions of each other. Further, we show that if $-\infty<\beta<\alpha<2$ and $0<q<p<\infty$ then they can be used to transform the L\'evy measures of $TS^p_\beta$ distributions into those of $TS^q_\alpha$.