Three Upsilon Transforms Related to Tempered Stable Distributions

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

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