Barndorff-Nielsen, Ole Eiler and Lindner, Alexander M.
Some aspects of Lévy copulas.
Collaborative Research Center 386, Discussion Paper 388
Levy processes and infinitely divisible distributions are increasingly defined in terms of their Levy measure. In order to describe the dependence structure of a multivariate Levy measure, Tankov (2003) introduced positive Levy copulas. Together with the marginal Levy measures they completely describe multivariate Levy measures on the first quadrant. In this paper, we show that any such Levy copula defines itself a Levy measure with 1-stable margins, in a canonical way. A limit theorem is obtained, characterising convergence of Levy measures with the aid of Levy copulas. Homogeneous Levy copulas are considered in detail. They correspond to Levy processes which have a time-constant Levy copula. Furthermore, we show how the Levy copula concept can be used to construct multivariat distributions in the Bondesson class with prescribed margins in the Bondesson class. The construction depends on a mapping Upsilon, recently introduced by Barndorff-Nielsen and Thorbjornsen (2004a,b) and Barndorff-Nielsen, Maejima and Sato (2004). Similar results are obtained for self-decomposable distributions and for distributions in the Thorin class.