Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations

Abstract

The class of distributions on $ℝ$ 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 $T\left({ℝ}^d\right)$ and $B\left({ℝ}^d\right)$ . From the Lévy process $\left\{X_t^{\left(\mu \right)}\right\}$ on ${ℝ}^d$ with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral ${\int }_0^{1}log(1/t)d{X}_t^{\left(\mu \right)}$ . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that $\Upsilon \left(\mathrm{ID}\right({ℝ}^d\left)\right)=B\left({ℝ}^d\right)$ and $\Upsilon \left(L\right({ℝ}^d\left)\right)=T\left({ℝ}^d\right)$ , where $\mathrm{ID}\left({ℝ}^d\right)$ and $L\left({ℝ}^d\right)$ are the classes of infinitely divisible distributions and of self-decomposable distributions on ${ℝ}^d$ , 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 $\left\{X_t^{\left(\mu \right)}\right\}$ are studied. Developments of these results in the context of the nested sequence $L_m\left({ℝ}^d\right)$, $m=0,1,\dots ,\mathrm{\infty }$ , are presented. Other applications and examples are given.