### Pareto Lévy measures and multivariate regular variation

#### Abstract

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

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

Dates
First available in Project Euclid: 8 March 2012

https://projecteuclid.org/euclid.aap/1331216647

Digital Object Identifier
doi:10.1239/aap/1331216647

Mathematical Reviews number (MathSciNet)
MR2951549

Zentralblatt MATH identifier
1248.60052

#### Citation

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. https://projecteuclid.org/euclid.aap/1331216647

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