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(Rd) and B(Rd) . From the Lévy process {Xt(μ)} on Rd with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral ∫01log(1/t)dXt(μ) . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that ϒ(ID(Rd))=B(Rd) and ϒ(L(Rd))=T(Rd) , where ID(Rd) and L(Rd) are the classes of infinitely divisible distributions and of self-decomposable distributions on Rd , 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 {Xt(μ)} are studied. Developments of these results in the context of the nested sequence Lm(Rd), m=0,1,…,∞ , are presented. Other applications and examples are given. Keywords: Goldie-Steutel-Bondesson class; infinite divisibility; Lévy measure; Lévy process; self-decomposability; stochastic integral; Thorin class

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