Asymptotic independence in higher dimensions and its implications on risk management

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 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. Notably,for the popular Gaussian copula, we provide explicit conditions on the correlation matrix for mutual asymptotic independence and k-wise asymptotic independence to hold; moreover, we are able to compute exact tail orders for various tail events. Beside that, we compare and discuss the implications of these new notions of asymptotic independence on assessing the risk of complex systems under distributional ambiguity.