Some aspects of Lévy copulas

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.