A criterion and a Cram\'er-Wold device for quasi-infinite divisibility for discrete multivariate probability laws

TL;DR

This paper characterizes quasi-infinite divisibility of multivariate discrete probability laws through their characteristic functions and extends univariate results to multivariate cases, providing new criteria and Cramér-Wold devices.

Contribution

It introduces a criterion based on characteristic functions for quasi-infinite divisibility and generalizes Cramér-Wold devices to multivariate laws.

Findings

Quasi-infinite divisibility is equivalent to characteristic functions being separated from zero.

Extended Cramér-Wold devices for infinite and quasi-infinite divisibility.

Generalized results from univariate to multivariate discrete laws.

Abstract

Multivariate discrete probability laws are considered. We show that such laws are quasi-infinitely divisible if and only if their characteristic functions are separated from zero. We generalize the existing results for the univariate discrete laws and for the multivariate laws on $\mathbb{Z}^d$. The Cram\'er-Wold devices for infinite and quasi-infinite divisibility were proved.