A quasi-infinitely divisible distribution on R d is a probability distribution µ on R d whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on R d . Equivalently, it can be characterised as a probability distribution whose characteristic function has a Lévy-Khintchine type representation with a "signed Lévy measure", a so called quasi-Lévy measure, rather than a Lévy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato [20]. The goal of the present paper is to collect some known results on multivariate quasiinfinitely 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 Z d -valued quasi-infinitely divisible distributions.
We consider distributions on
R
\mathbb {R}
that can be written as the sum of a non-zero discrete distribution and an absolutely continuous distribution. We show that such a distribution is quasi-infinitely divisible if and only if its characteristic function is bounded away from zero, thus giving a new class of quasi-infinitely divisible distributions. Moreover, for this class of distributions we characterize the existence of the
g
g
-moment for certain functions
g
g
.
A probability distribution µ on R d is quasi-infinitely divisible if its characteristic function has the representation µ = µ 1 / µ 2 with infinitely divisible distributions µ 1 and µ 2 . In [6, Thm. 4.1] it was shown that the class of quasi-infinitely divisible distributions on R is dense in the class of distributions on R with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on R d is not dense in the class of distributions on R d with respect to weak convergence if d ≥ 2.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.