Advances in Applied Probability

Pareto Lévy measures and multivariate regular variation

Irmingard Eder and Claudia Klüppelberg

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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.

Article information

Adv. in Appl. Probab., Volume 44, Number 1 (2012), 117-138.

First available in Project Euclid: 8 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; Lévy processes 60G52: Stable processes
Secondary: 60G70: Extreme value theory; extremal processes

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


Eder, Irmingard; Klüppelberg, Claudia. Pareto Lévy measures and multivariate regular variation. Adv. in Appl. Probab. 44 (2012), no. 1, 117--138. doi:10.1239/aap/1331216647.

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