Metric embeddings of tail correlation matrices

Abstract
The assessment of risks associated with multivariate random vectors relies heavily on understanding their extremal dependence, crucial in evaluating risk measures for financial or insurance portfolios. A widely used metric for assessing tail risk is the tail correlation matrix of tail correlation coefficients. Among the exploration of structural properties of the tail correlation matrix the so-called realization problem of deciding whether a given matrix is the tail correlation matrix of some underlying random vector has recently received some attention.

The entries of the tail correlation matrix are closely related to a useful distance measure on the space of Frechet-random variables, named spectral distance and first introduced in Davis & Resnick (1993). We analyze the properties of a related semimetric and show that it has the special property of being embeddable both in vector and function space, equipped with the respective sum norm. Notably, these embeddings bear a direct relationship with the realization of specific tail dependence structures via max-stable random vectors. Particularly, an embedding in vector space, employing so-called line metrics, provides a representation through a max-stable mixture of so-called Tawn-Molchanov models, s. also Fiebig, Strokorb & Schlather (2017).

Leveraging this framework, we revisit the realization problem, affirming a conjecture by Shyamalkumar & Tao (2020) regarding its NP-completeness.

This talk is based on joint work with Sebastian Neblung (University of Hamburg) and Stilian Stoev (University of Michigan).