On multivariate quasi-infinitely divisible distributions

TL;DR

This paper reviews and extends the theory of multivariate quasi-infinitely divisible distributions, focusing on their properties, convergence, moments, support, and examples, especially those valued in  lattices.

Contribution

It generalizes univariate results to the multivariate case, providing conditions and examples for these distributions, including -valued cases.

Findings

Conditions for weak convergence established

Moment and support properties analyzed

Examples of -valued quasi-infinitely divisible distributions provided

Abstract

A quasi-infinitely divisible distribution on $\mathbb{R}^d$ is a probability distribution $\mu$ on $\mathbb{R}^d$ whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on $\mathbb{R}^d$. Equivalently, it can be characterised as a probability distribution whose characteristic function has a L\'evy--Khintchine type representation with a "signed L\'evy measure", a so called quasi--L\'evy measure, rather than a L\'evy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato \cite{lindner}. The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on $\mathbb{Z}^d$-valued quasi-infinitely divisible distributions.