Asymptotic independence in more than two dimensions and its implications on risk management

In extreme value theory, the presence of asymptotic independence signifies that joint extreme events across multiple variables are unlikely. Although well understood in a bivariate context, the concept remains relatively unexplored when addressing the nuances of simultaneous occurrence of extremes in higher dimensions. In this article, we propose a notion of mutual asymptotic independence to capture the behaviour of joint extremes in dimensions larger than two and contrast it with the classical notion of (pairwise) asymptotic independence. Additionally, we define -wise asymptotic independence, which captures the tail dependence in 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 in complex systems under distributional ambiguity.