A Sufficient Condition for Absolute Continuity of Infinitely Divisible Distributions

TL;DR

This paper establishes a new sufficient condition involving the Lévy measure for the absolute continuity of symmetric infinitely divisible distributions, expanding the understanding beyond existing criteria.

Contribution

It introduces a novel sufficient condition based on the Lévy measure that guarantees absolute continuity, not implied by prior results.

Findings

The condition               

The result is not implied by existing theorems about absolute continuity.

Provides a new criterion involving the Lévy measure for symmetric infinitely divisible distributions.

Abstract

We consider infinitely divisible distributions with symmetric L\'evy measure and study the absolute continuity of them with respect to the Lebesgue measure. We prove that if $\eta(r)=\int_{|x|\le r} x^2 \nu(dx)$ where $\nu$ is the L\'evy measure, then $\int_0^1 \frac{r}{\eta(r)}dr <\infty$ is a sufficient condition for absolute continuity. As far as we know, our result is not implied by existing results about absolute continuity of infinitely divisible distributions.