Type \(G\) distributions on \({\mathbb R}^d\)
The authors present a systematic study of the class of multivariate distributions obtained by a Gaussian randomization of jumps of a Lévy process. It is shown that this class of distributions, called type \(G\) distributions, is a closed convolution semigroup of the set of symmetric infinitely divisible measures with respect to weak convergence. The Lévy measure of a type \(G\) distribution is characterized in terms of a polar decomposition and so called type \(G\) processes are introduced as those with type \(G\) finite-dimensional marginals. The authors construct an infinitely divisible distribution which is not of type \(G\), but all lower dimensional projections, especially the one-dimensional marginals, are of type \(G\). Moreover, a series representation of type \(G\) random vectors is presented.
scientific article; zbMATH DE number 1784211