In this paper we study two important representations for extreme value distributions and their max-domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so-called auxiliary functions psi. Up to now, however, the set of valid auxiliary functions for vMR has neither been characterized completely nor separated from those for VR. We contribute to the current literature by introducing ''universal'' auxiliary functions which are valid for both VR and vMR representations for the entire MDA distribution families. Then we identify exactly the sets of valid auxiliary functions for both VR and vMR. Moreover, we propose a method for finding appropriate auxiliary functions with analytically simple structure and provide them for several important distributions.

翻译：本文研究极值分布及其最大吸引域（MDA）的两种重要表征——冯·米塞斯表征（vMR）与变分表征（VR），二者均为推导极限结果的便捷途径。VR与vMR均通过所谓的辅助函数psi定义。然而，迄今为止，vMR的有效辅助函数集合既未得到完整刻画，也未与VR的辅助函数集合明确区分。本文通过引入适用于整个MDA分布族中VR与vMR表征的"通用"辅助函数，为现有文献做出贡献。进而精确识别VR与vMR各自的有效辅助函数集合。此外，我们提出一种构造具有解析简单结构的合适辅助函数的方法，并为若干重要分布给出了相应函数。