Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations

Abstract

The class of distributions on $\\mathbb{R}$ 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 $T(\\mathbb{R}^d)$ and $B(\\mathbb{R}^d)$ . From the Lévy process $\\{X_t^{(\\mu)}\\}$ on $\\mathbb{R}^d$ with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral $\\int_0^1 \\log(1/t)\\d X_t^{(\\mu)}$ . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that $\\Upsilon(ID(\\mathbb{R}^d))=B(\\mathbb{R}^d)$ and $\\Upsilon(L(\\mathbb{R}^d))=T(\\mathbb{R}^d)$ , where $ID(\\mathbb{R}^d)$ and $L(\\mathbb{R}^d)$ are the classes of infinitely divisible distributions and of self-decomposable distributions on $\\mathbb{R}^d$ , 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 $\\{X_t^{(\\mu)}\\}$ are studied. Developments of these results in the context of the nested sequence $L_m(\\mathbb{R}^d)$, $m=0,1,\\ldots,\\infty$ , [math] , are presented. Other applications and examples are given.