March 2012 Pareto Lévy measures and multivariate regular variation
Irmingard Eder, Claudia Klüppelberg
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Adv. in Appl. Probab. 44(1): 117-138 (March 2012). DOI: 10.1239/aap/1331216647


We consider regular variation of a Lévy process X := (X_t)t≥0 in Rd with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.


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Irmingard Eder. Claudia Klüppelberg. "Pareto Lévy measures and multivariate regular variation." Adv. in Appl. Probab. 44 (1) 117 - 138, March 2012.


Published: March 2012
First available in Project Euclid: 8 March 2012

zbMATH: 1248.60052
MathSciNet: MR2951549
Digital Object Identifier: 10.1239/aap/1331216647

Primary: 60E07 , 60G51 , 60G52
Secondary: 60G70

Keywords: Dependence of Lévy processes , Lévy copula , Lévy measure , multivariate regular variation , multivariate stable process , Pareto Lévy copula , spectral measure , tail dependence coefficient , tail integral

Rights: Copyright © 2012 Applied Probability Trust


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Vol.44 • No. 1 • March 2012
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