A class of multivariate infinitely divisible distributions related to arcsine density

TL;DR

This paper introduces a new class of multivariate infinitely divisible distributions related to arcsine density, characterizes their Lévy measures, and explores their relation to Upsilon transformations and generalized type G distributions.

Contribution

It defines the class A of distributions with Lévy measures derived from arcsine-based transformations and links them to stochastic integrals and existing distribution classes.

Findings

The class A includes the Jurek class as a proper subset.

The transformation $\\mathcal{A}_2$ is an Upsilon transformation.

The transformation $\mathcal{A}_1$ is not an Upsilon transformation.

Abstract

Two transformations $\mathcal{A}_1$ and $\mathcal{A}_2$ of L\'{e}vy measures on $\mathbb{R}^d$ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of $\mathcal{A}_1$ and $\mathcal{A}_2$ are determined and it is shown that they have the same range. The class of infinitely divisible distributions on $\mathbb{R}^d$ with L\'{e}vy measures being in the common range is called the class $A$ and any distribution in the class $A$ is expressed as the law of a stochastic integral $\int_0^1\cos(2^{-1}\uppi t)\,\mathrm{d}X_t$ with respect to a L\'{e}vy process $\{X_t\}$. This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type $G$ distributions are the image of distributions in the class $A$ under a mapping defined by an appropriate stochastic integral. $\mathcal{A}_2$ is identified as an Upsilon transformation, while $\mathcal{A}_1$ is shown not to be.