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. Furthermore, we define $k$-wise asymptotic independence which lies in 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 to hold; moreover, we are able to compute exact tail orders for various tail events.

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。尽管在二元情形下已有深入理解，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探索。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，我们定义了介于成对渐近独立与互渐近独立之间的k重渐近独立性。通过阿基米德族、高斯族及Marshall-Olkin族等多种连接函数的实例对这些概念进行了比较。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的显式条件；此外，我们能够计算各类尾事件的精确尾部阶数。