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Sparse regular variation

Part of: Multivariate analysis Stochastic processes

Published online by Cambridge University Press:  22 November 2021

Nicolas Meyer Open the ORCID record for Nicolas Meyer [Opens in a new window]  and
Olivier Wintenberger
Nicolas Meyer*
Affiliation:
Sorbonne Université
Olivier Wintenberger*
Affiliation:
Sorbonne Université
*
*Postal address: Sorbonne Université, LPSM, 4 place Jussieu, F-75005, Paris, France.
*Postal address: Sorbonne Université, LPSM, 4 place Jussieu, F-75005, Paris, France.
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Abstract

Regular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

Keywords

Multivariate extremesregular variationspectral measure

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62H05: Characterization and structure theory
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1115 - 1148
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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