Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes

TL;DR

This paper explores the inversion of Levy measures to connect the short and long-term behaviors of Levy processes, providing conditions for convergence to stable distributions at both time scales.

Contribution

It generalizes the concept of Levy measure inversion and applies it to establish relationships between process behaviors as time approaches zero and infinity.

Findings

Derived conditions for Levy process convergence to stable distributions at zero and infinity

Established relationships for tempered stable Levy processes using Rosinski measures

Provided a unified framework for analyzing Levy process asymptotics

Abstract

The inversion of a Levy measure was first introduced (under a different name) in Sato 2007. We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Levy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Levy process as time approaches infinity. This allows us to get self contained conditions for a Levy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Levy processes and for the important class of tempered stable Levy processes. For this latter class, we give detailed results in terms of their Rosinski measures.