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Linking representations for multivariate extremes via a limit set

Published online by Cambridge University Press:  13 June 2022

Natalia Nolde  and
Jennifer L. Wadsworth
Natalia Nolde*
Affiliation:
University of British Columbia
Jennifer L. Wadsworth*
Affiliation:
Lancaster University
*
*Postal address: Department of Statistics, 3182 Earth Sciences Building, 2207 Main Mall, Vancouver, BC, V6T 1Z4, Canada. Email address: [email protected]
**Postal address: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, LA1 4YF, UK. Email address: [email protected]
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Abstract

The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.

Keywords

Multivariate extreme value theoryconditional extremeshidden regular variationlimit setasymptotic (in)dependence
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 688 - 717
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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