Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator

We provide a proof that the empirical spatial distribution estimator in $\mathbb R^d$ as well as the corresponding plug-in estimator of the spatial depth are uniformly $L^1$-consistent.

The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters.

This is a rare property.

The result of this note originates from a conversation with ChatGPT 5.4 Pro as part of some of our own earlier experiments on its mathematical reasoning capabilities.