The class of distributions on generated by convolutions of Γ-distributions and the class generated by convolutions of mixtures of exponential distributions are generalized to higher dimensions and denoted by and . From the Lévy process on with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that and , where and are the classes of infinitely divisible distributions and of self-decomposable distributions on , respectively. The relations with the mapping Φ from μ to the distribution at each time of the stationary process of Ornstein-Uhlenbeck type with background driving Lévy process are studied. Developments of these results in the context of the nested sequence , , are presented. Other applications and examples are given.
"Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations." Bernoulli 12 (1) 1 - 33, February 2006.