On multivariate quasi-infinitely divisible distributions

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.