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 $L_m(\mathbb{R}^d)$, $m=0,1,\ldots,\infty$, , are presented. Other applications and examples are given.