Linking representations for multivariate extremes via a limit set

TL;DR

This paper explores the connections between various models of multivariate extreme dependence by using a geometric approach centered on the shape of the limit set of scaled sample clouds.

Contribution

It introduces a unifying geometric framework that links different extremal dependence representations through the shape of the limit set.

Findings

Limit set shape characterizes extremal dependence models

Connections between hidden regular variation and conditional models are clarified

Geometric approach provides new insights into multivariate extremes

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 exploring what happens when extremes of one variable may grow at different rates to 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.