  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.
