Subexponential densities of infinitely divisible distributions on the half line

TL;DR

This paper establishes conditions under which the densities of infinitely divisible distributions on the half line are subexponential, linking their properties to those of their normalized Lévy measures, with implications for understanding tail behaviors.

Contribution

It provides a characterization of subexponential densities of infinitely divisible distributions based on properties of their normalized Lévy measures, under long-tailedness and continuity assumptions.

Findings

Densities are subexponential iff normalized Lévy measures are subexponential under long-tailedness.

Densities are subexponential iff normalized Lévy measures are locally subexponential under continuity.

Results connect tail behavior of distributions to properties of Lévy measures.

Abstract

We show that, under the long-tailedness of the densities of normalized L\'evy measures, the densities of infinitely divisible distributions on the half line are subexponential if and only if the densities of their normalized L\'evy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half line are subexponential if and only if their normalized L\'evy measures are locally subexponential.