On the convolution equivalence of tempered stable distributions on the real line

TL;DR

This paper establishes the convolution equivalence property of univariate tempered stable distributions, providing a rigorous foundation for their asymptotic similarity in probability and Lévy densities, with illustrative examples.

Contribution

It proves the convolution equivalence property for univariate tempered stable distributions, clarifying their asymptotic behavior and connecting heuristic arguments with rigorous results.

Findings

Confirmed convolution equivalence for tempered stable distributions

Clarified asymptotic similarity between probability and Lévy densities

Discussed specific examples from existing literature

Abstract

We show the convolution equivalence property of univariate tempered stable distributions in the sense of Rosi\'nsky (2007). This makes rigorous various classic heuristic arguments on the asymptotic similarity between the probability and L\'evy densities of such distributions. Some specific examples from the literature are discussed.