Dependence of multivariate extremes

TL;DR

This paper establishes precise conditions under which sub-vectors of multivariate extreme value distributions are independent or totally dependent, linking extremal indexes and dependence coefficients, with practical illustrations.

Contribution

It provides necessary and sufficient conditions for independence and dependence of sub-vectors in multivariate extreme value distributions, involving extremal indexes and dependence coefficients.

Findings

Conditions for independence and dependence are characterized mathematically.

Illustrations include auto-regressive and 3-dependent sequences.

Results enhance understanding of dependence structures in multivariate extremes.

Abstract

We give necessary and sufficient conditions for two sub-vectors of a random vector with a multivariate extreme value distribution, corresponding to the limit distribution of the maximum of a multidimensional stationary sequence with extremal index, to be independent or totally dependent. Those conditions involve first relations between the multivariate extremal indexes of the sequences and secondly a coefficient that measure the strength of dependence between both sub-vectors. The main results are illustrated with an auto-regressive sequence and a 3-dependent sequence.