Inversion formula for infinitely divisible distributions

TL;DR

This paper proves an inversion formula that allows the computation of the Lévy measure of an infinitely divisible distribution directly from its characteristic function, paralleling classical inversion formulas for distribution functions.

Contribution

It introduces a new inversion formula for Lévy measures of infinitely divisible distributions based on their characteristic functions.

Findings

Provides a method to compute Lévy measures from characteristic functions

Establishes a formula similar to classical distribution inversion formulas

Facilitates analysis of infinitely divisible distributions

Abstract

The aim of this note is to prove the inversion formula, which can be used to compute the Levi measure of an infinitely divisible distribution from its characteristic function. Obtained formula is similar to the well-known inversion formula [2], which is used to compute the distribution function of a random variable from the corresponding characteristic function.