Lévy Copulas: Dynamics and Transforms of Upsilon Type
Lévy processes and infinitely divisible distributions are increasingly defined in terms of their Lévy measure. In order to describe the dependence structure of a multivariate Lévy measure, Tankov (2003) introduced Lévy copulas on <formula format="inline"><file name="sjos_527_mu1.gif" type="gif" /></formula>. (For an extension to <b><openface>R</openface>-super-<b>""m""</b></b>, see Kallsen & Tankov, 2006.) Together with the marginal Lévy measures they completely describe multivariate Lévy measures on <formula format="inline"><file name="sjos_527_mu2.gif" type="gif" /></formula>. In this article we show that any such Lévy copula defines itself a Lévy measure with one-stable margins, in a canonical way. A limit theorem is obtained, characterizing convergence of Lévy measures with the aid of Lévy copulas. Homogeneous Lévy copulas are considered in detail. They correspond to Lévy processes which have a time-constant Lévy copula, and a complete description of homogeneous Lévy copulas is obtained. A general scheme to construct multivariate distributions having special properties is outlined, for distributions with prescribed margins having the same properties. This makes use of Lévy copulas and of certain mappings of Upsilon type. The construction is then exemplified for distributions in the Goldie-Steutel-Bondesson class, the Thorin class and for self-decomposable distributions. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..
Year of publication: |
2007
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Authors: | BARNDORFF-NIELSEN, OLE E. ; LINDNER, ALEXANDER M. |
Published in: |
Scandinavian Journal of Statistics. - Danish Society for Theoretical Statistics, ISSN 0303-6898. - Vol. 34.2007, 2, p. 298-316
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Publisher: |
Danish Society for Theoretical Statistics Finnish Statistical Society Norwegian Statistical Association Swedish Statistical Association |
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