We study empirical variants of the halfspace (Tukey) depth of a probability measure $\mu$, which are obtained by replacing $\mu$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of $\mu$ being the uniform distribution on a convex body $K$, the depth trimmed regions are convex floating bodies of $K$, and we obtain strong limit theorems for their empirical estimators.

翻译：我们研究了概率测度 $\mu$ 的半空间（Tukey）深度的经验变体，这些变体是通过用相应的加权经验测度替换 $\mu$ 而得到的。我们证明了马钦凯维奇-齐格蒙强大数定律和重对数律在集合包含关系以及深度截断区域的理论与经验变体之间的豪斯多夫距离上的类似结果。在 $\mu$ 为凸体 $K$ 上均匀分布的特殊情况下，深度截断区域是 $K$ 的凸浮体，并且我们得到了其经验估计量的强极限定理。