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The Pareto Copula, Aggregation of Risks, and the Emperor's Socks

Part of: Nonparametric inference Limit theorems Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Claudia Klüppelberg  and
Sidney I. Resnick
Claudia Klüppelberg*
Affiliation:
Munich University of Technology
Sidney I. Resnick*
Affiliation:
Cornell University
*
∗Postal address: Chair of Mathematical Statistics, Center for Mathematical Sciences, Munich University of Technology, Boltzmannstrasse 3, 85747 Garching bei München, Germany. Email address: cklu@ma.tum.de
∗∗Postal address: School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: sir1@cornell.edu
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Abstract

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The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.

Keywords

Regular variationriskmaximal domain of attractioncopulaPareto

MSC classification

Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60G70: Extreme value theory; extremal processes 60F17: Functional limit theorems; invariance principles 62G30: Order statistics; empirical distribution functions
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 67 - 84
Copyright
Copyright © Applied Probability Trust 2008 

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