Strong limit theorems for empirical halfspace depth trimmed regions

TL;DR

This paper establishes strong limit theorems for empirical halfspace depth trimmed regions, including laws of large numbers and iterated logarithm results, with applications to convex floating bodies.

Contribution

It provides the first strong limit theorems for empirical halfspace depth trimmed regions, extending classical probabilistic results to geometric depth regions.

Findings

Proves strong law of large numbers for depth regions

Establishes law of the iterated logarithm for depth regions

Applies results to convex floating bodies of uniform distributions

Abstract

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.