Inversions of infinitely divisible distributions and conjugates of stochastic integral mappings

TL;DR

This paper explores the mathematical properties of inversions and conjugates of infinitely divisible distributions, introducing new concepts and characterizations related to stochastic integral mappings and their iterative limits.

Contribution

It introduces the concept of conjugates for stochastic integral mappings and characterizes their domains and ranges, linking them to inversions of distributions.

Findings

Properties and characterizations of inversions are provided.

Domains and ranges of conjugates for specific stochastic integral mappings are described.

Applications to the limits of iterative stochastic integral mappings are discussed.

Abstract

The dual of an infinitely divisible distribution on $\mathbb{R}^d$ without Gaussian part defined in Sato, ALEA {\bf 3} (2007), 67--110, is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping $\mu=\Phi_{f}\rho$ of $\rho$ to $\mu$ in the class of infinitely divisible distributions on $\mathbb{R}^d$, where $\mu$ is the distribution of an improper stochastic integral of a nonrandom function $f$ with respect to a L\'{e}vy process on $\mathbb{R}^d$ with distribution $\rho$ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.