Tail Properties of Multivariate Archimedean Copulas

TL;DR

This thesis explores the tail behavior of multivariate Archimedean copulas using probabilistic representations, introducing new asymptotic results and providing clearer insights into tail dependence, extreme value limits, and threshold copulas.

Contribution

It introduces new asymptotic results for the Williamson d-transform, simplifying the analysis of tail properties of Archimedean copulas through probabilistic methods.

Findings

Derived new results on the asymptotic behavior of the Williamson d-transform.

Provided transparent derivations of tail dependence coefficients and extreme value limits.

Enhanced understanding of the probabilistic structure underlying Archimedean copulas.

Abstract

In this thesis, the tail properties of multivariate Archimedean copulas are investigated using known representation theorems involving L1-norm symmetric distributions and the Williamson d-transform. Several new results on the asymptotic properties of the Williamson d-transform are established and subsequently used to study the tails of Archimedean copulas. This makes it possible to recover many known results regarding their tail behavior in a straightforward and transparent way. In particular, coefficients of tail dependence, extreme value limits and threshold copulas are considered. A central theme is the emphasis on the probabilistic aspects of stochastic representations, rather than the analytic aspects of representations involving Archimedean generators.