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：我们研究概率测度 $\mu$ 的半空间（Tukey）深度的经验变体，这些变体通过将 $\mu$ 替换为相应的加权经验测度而获得。我们证明了Marcinkiewicz--Zygmund强大数定律和迭代对数定律的类似物，涉及深度修剪区域的理论变体与经验变体之间的集合包含关系以及Hausdorff距离。在 $\mu$ 是凸体 $K$ 上均匀分布的特殊情形下，深度修剪区域即为 $K$ 的凸浮动体，我们得到了其经验估计量的强极限定理。