On the convolution equivalence of tempered stable distributions on the real line
Lorenzo Torricelli
TL;DR
This paper proves that univariate tempered stable distributions on the real line, defined via completely monotone tempering of their Lévy density, are convolution equivalent, linking the tails of the distribution and its Lévy measure. By constructing a normalized Lévy density $\nu_1$ and showing it lies in the class $\mathcal L(\gamma)$ and $\mathcal S(\gamma)$ with $\gamma=\inf\text{supp}(Q_+)$, the authors derive precise tail asymptotics for the pdf: $p(x)\sim \delta_\pm \hat{\mu}_q^\pm(\gamma_\pm)\frac{q_\pm(|x|)}{|x|^{1+\alpha}}$ as $x\to\pm\infty$. This yields an explicit proportionality between the tails of the distribution and its Lévy density, and provides concrete asymptotic results for various tempering schemes, including exponential, KR, and Mittag-Leffler tempering. The Gamma distribution is highlighted as a boundary case that is not TS$_\alpha$, illustrating the sharpness of the results. The findings have practical impact for modeling heavy-tailed phenomena in finance and related fields by enabling rigorous tail approximations and moment control via tempering.
Abstract
We show the convolution equivalence property of univariate tempered stable distributions in the sense of Rosińsky (2007). This makes rigorous various classic heuristic arguments on the asymptotic similarity between the probability and Lévy densities of such distributions. Some specific examples from the literature are discussed.