There are many ways of measuring and modeling tail-dependence in random vectors: from the general framework of multivariate regular variation and the flexible class of max-stable vectors down to simple and concise summary measures like the matrix of bivariate tail-dependence coefficients. This paper starts by providing a review of existing results from a unifying perspective, which highlights connections between extreme value theory and the theory of cuts and metrics. Our approach leads to some new findings in both areas with some applications to current topics in risk management.

We begin by using the framework of multivariate regular variation to show that extremal coefficients, or equivalently, the higher-order tail-dependence coefficients of a random vector can simply be understood in terms of random exceedance sets, which allows us to extend the notion of Bernoulli compatibility. In the special but important case of bivariate tail-dependence, we establish a correspondence between tail-dependence matrices and \(L^1\)- and \(\ell _1\)-embeddable finite metric spaces via the spectral distance, which is a metric on the space of jointly 1-Fréchet random variables. Namely, the coefficients of the cut-decomposition of the spectral distance and of the Tawn-Molchanov max-stable model realizing the corresponding bivariate extremal dependence coincide. We show that line metrics are rigid and if the spectral distance corresponds to a line metric, the higher order tail-dependence is determined by the bivariate tail-dependence matrix.

Finally, the correspondence between \(\ell _1\)-embeddable metric spaces and tail-dependence matrices allows us to revisit the realizability problem, i.e. checking whether a given matrix is a valid tail-dependence matrix. We confirm a conjecture of Shyamalkumar and Tao (2020) that this problem is NP-complete.

测量和建模随机向量中的尾部依赖性的方法有很多：从多元正则变异的一般框架和最大稳定向量的灵活类别到简单而简洁的汇总测量，例如二元尾部依赖性系数矩阵。本文首先从统一的角度对现有结果进行回顾，强调极值理论与削减和度量理论之间的联系。我们的方法在这两个领域带来了一些新的发现，并在当前的风险管理主题中得到了一些应用。

我们首先使用多元正则变化的框架来证明极值系数，或者等效地，随机向量的高阶尾部相关系数可以简单地用随机超越集来理解，这使我们能够扩展以下概念：伯努利相容性。在双变量尾部依赖的特殊但重要的情况下，我们通过谱距离建立尾部依赖矩阵与\(L^1\) - 和\(\ell _1\) -可嵌入有限度量空间之间的对应关系，即联合 1-Fréchet 随机变量空间的度量。即，光谱距离的割分解的系数与实现相应的二元极值依赖性的Tawn-Molchanov最大稳定模型的系数一致。我们表明，线度量是刚性的，如果谱距离对应于线度量，则高阶尾部依赖性由二元尾部依赖性矩阵确定。

最后，可嵌入度量空间和尾部依赖矩阵之间的对应关系使我们能够重新审视可实现性问题，即检查给定矩阵是否是有效的尾部依赖矩阵。我们证实了Shyamalkumar和Tao（2020）的猜想，即这个问题是NP完全的。