Tempered geometric stable distributions and processes

TL;DR

This paper introduces a new class of tempered geometric stable distributions using Mittag-Leffler functions, analyzing their properties, spectral densities, and scaling limits of related Lévy processes.

Contribution

It develops a novel geometric tempering approach with Mittag-Leffler functions and thoroughly investigates the properties and scaling behaviors of the resulting distributions and processes.

Findings

Derived characteristic exponents and cumulants

Established absolute continuity relations

Analyzed short and long time scaling limits

Abstract

We introduce a notion of geometric tempering using exponentially-dampened Mittag-Leffler tempering functions and closely investigate the univariate case. Characteristic exponents and cumulants are calculated, as well as spectral densities. Absolute continuity relations are shown, and short and long time scaling limits of the associated L\'evy processes analyzed.