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.

翻译：在极值研究中，渐近独立性的存在意味着多个变量的极端事件同时发生的可能性较低。尽管在二元情境下已有充分理解，但在处理高维联合极端事件的细微特征时，这一概念仍相对缺乏深入探讨。本文提出互渐近独立性的概念，以刻画二维以上联合极端事件的行为特征，并将其与经典的（成对）渐近独立性概念进行对比。此外，我们定义了k阶渐近独立性，用以描述介于成对渐近独立性与互渐近独立性之间的尾部相依关系。通过阿基米德、高斯及马歇尔-奥金等多种连接函数的示例，对这些概念进行了比较分析。特别地，针对广泛应用的高斯连接函数，我们给出了相关矩阵满足互渐近独立性与k阶渐近独立性的显式条件，并能够计算各类尾部事件的精确尾部阶数。在此基础上，我们进一步比较并讨论了这些新提出的渐近独立性概念在分布模糊性下评估复杂系统风险方面的应用价值。