The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. We provide a functional asymptotic expansion for the empirical angular measure in the bivariate case based on the theory of weak convergence in the space of bounded functions. From the expansion, not only can the known asymptotic distribution of the empirical angular measure be recovered, it also enables to find expansions and weak limits for other statistics based on the associated empirical process or its quantile version.

翻译：单位球面上的角测度刻画了随机向量各分量在极值区域的一阶依赖结构，其定义基于标准化边缘分布。该测度的统计重构是处理远离数据中心观测值的学习问题中的重要步骤。当向量各分量具有不同分布时，秩变换为数据标准化提供了便捷而稳健的方法，从而能够基于最极端的观测值构建角测度的经验版本。本文基于有界函数空间弱收敛理论，给出了二元情形下经验角测度的泛函渐近展开。通过该展开式，不仅可以还原经验角测度的已知渐近分布，还能推导基于相关经验过程或其分位数版本的其他统计量的展开式与弱极限。