We consider the empirical versions of geometric quantile and halfspace depth, and study their extremal behaviour as a function of the sample size. The objective of this study is to establish connection between the rates of convergence and tail behaviour of the corresponding underlying distributions. The intricate interplay between the sample size and the parameter driving the extremal behaviour forms the main result of this analysis. In the process, we also fill certain gaps in the understanding of population versions of geometric quantile and halfspace depth.

翻译：我们考虑几何分位数和半空间深度的经验版本，并研究它们随样本量变化的极值行为。本研究的目的是建立收敛速率与相应基础分布尾部行为之间的联系。样本量与驱动极值行为的参数之间的复杂相互作用构成了本分析的主要结果。在此过程中，我们还填补了关于几何分位数和半空间深度总体版本理解中的某些空白。