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Pareto Lévy Measures and Multivariate Regular Variation

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Irmingard Eder  and
Claudia Klüppelberg
Irmingard Eder*
Affiliation:
Technische Universität München
Claudia Klüppelberg*
Affiliation:
Technische Universität München
*
∗ Postal address: Centre for Mathematical Sciences, and Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany.
∗ Postal address: Centre for Mathematical Sciences, and Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany.
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Abstract

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We consider regular variation of a Lévy process X := (Xt)t≥0 in 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.

Keywords

Dependence of Lévy processesLévy copulaLévy measurePareto Lévy copulamultivariate regular variationmultivariate stable processspectral measuretail integraltail dependence coefficient

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 117 - 138
Copyright
© Applied Probability Trust 

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