  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.
