A criterion and a Cramér-Wold device for quasi-infinite divisibility for discrete multivariate probability laws

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 Cramér-Wold devices for infinite and quasi-infinite divisibility were proved.