We develop an asymptotic theory for extremes in decomposable graphical models by presenting results applicable to a range of extremal dependence types. Specifically, we investigate the weak limit of the distribution of suitably normalised random vectors, conditioning on an extreme component, where the conditional independence relationships of the random vector are described by a chordal graph. Under mild assumptions, the random vector corresponding to the distribution in the weak limit, termed the tail graphical model, inherits the graphical structure of the original chordal graph. Our theory is applicable to a wide range of decomposable graphical models including asymptotically dependent and asymptotically independent graphical models. Additionally, we analyze combinations of copula classes with differing extremal dependence in cases where a normalization in terms of the conditioning variable is not guaranteed by our assumptions. We show that, in a block graph, the distribution of the random vector normalized in terms of the random variables associated with the separators converges weakly to a distribution we term tail noise. In particular, we investigate the limit of the normalized random vectors where the clique distributions belong to two widely used copula classes, the Gaussian copula and the max-stable copula.

翻译：我们通过提出适用于多种极值依赖类型的结果，为可分解图模型建立了极值的渐近理论。具体而言，我们研究了在给定极端分量条件下经过适当归一化的随机向量分布的弱极限，其中随机向量的条件独立关系由弦图描述。在温和假设下，对应于弱极限分布的随机向量（称为尾部图模型）继承了原始弦图的图结构。该理论适用于广泛的可分解图模型，包括渐近依赖与渐近独立的图模型。此外，我们分析了协方差类在不同极值依赖组合下的情况，其中我们的假设无法保证基于条件变量进行归一化。研究表明，在块图中，基于分割变量相关联的随机变量进行归一化后的随机向量分布弱收敛于我们称之为“尾部噪声”的分布。特别地，我们研究了归一化随机向量的极限形式，其中团分布属于两种广泛使用的协方差类——高斯协方差与最大稳定协方差。