A key aspect where extreme values methods differ from standard statistical models is through having asymptotic theory to provide a theoretical justification for the nature of the models used for extrapolation. In multivariate extremes many different asymptotic theories have been proposed, partly as a consequence of the lack of ordering property with vector random variables. One class of multivariate models, based on conditional limit theory as one variable becomes extreme, developed by Heffernan and Tawn (2004), has developed wide practical usage. The underpinning value of this approach has been supported by further theoretical characterisations of the limiting relationships by Heffernan and Resnick (2007) and Resnick and Zeber (2014). However Drees and Jan{\ss}en (2017) provided a number of counterexamples of their results, which potentially undermine the trust in these statistical methods. Here we show that in the Heffernan and Tawn (2004) framework, which involves marginal standardisation to a common exponentially decaying tailed marginal distribution, the problems in these examples are removed.

翻译：极值方法与传统统计模型的一个关键区别在于，它利用渐近理论为外推模型的性质提供理论依据。在多变量极值中，由于向量随机变量缺乏排序性质，研究者提出了多种不同的渐近理论。基于Heffernan与Tawn（2004）发展的条件极限理论（当一个变量趋于极端时）的一类多变量模型，已获得广泛的实际应用。该方法的理论基础价值得到了Heffernan与Resnick（2007）以及Resnick与Zeber（2014）对极限关系的进一步理论刻画的支持。然而，Drees与Janßen（2017）提出了他们结果的一系列反例，这有可能削弱对这些统计方法的信任。本文证明，在Heffernan与Tawn（2004）的框架中，通过对边缘分布进行标准化使其服从共同指数衰减尾部的边缘分布，这些反例中的问题得以消除。