Asymptotic independence in higher dimensions and its implications on risk management

TL;DR

This paper introduces new concepts of asymptotic independence in higher dimensions, extending classical ideas to better understand joint extremes and improve risk assessment in complex systems.

Contribution

It proposes mutual and k-wise asymptotic independence notions, contrasting them with classical pairwise independence, and explores their implications for risk management.

Findings

New definitions of higher-dimensional asymptotic independence.

Comparison of concepts using Archimedean, Gaussian, and Marshall-Olkin copulas.

Implications for risk assessment under distributional ambiguity.

Abstract

In the study of extremes, the presence of asymptotic independence signifies that extreme events across multiple variables are probably less likely to occur together. Although well-understood in a bivariate context, the concept remains relatively unexplored when addressing the nuances of the joint occurrence of extremes in higher dimensions. In this paper, we propose a notion of mutual asymptotic independence to capture the behavior of joint extremes in dimensions larger than two and contrast it with the classical notion of (pairwise) asymptotic independence. Additionally, we define k-wise asymptotic independence, which captures the tail dependence between pairwise and mutual asymptotic independence. The concepts are compared using examples of Archimedean, Gaussian and Marshall-Olkin copulas, among others. Finally, we discuss the implications of these new notions of asymptotic independence on assessing the risk of complex systems under distributional ambiguity.