Three Upsilon Transforms Related to Tempered Stable Distributions

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 $TS^q_\alpha$.