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

Felix Gnettner; Hyemin Yeon; Piotr Kokoszka

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

Statistics Theory 2026-07-17 v1 Probability Machine Learning

Authors: Felix Gnettner , Hyemin Yeon , Piotr Kokoszka

Abstract

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.

Keywords

empirical spatial distribution spatial depth uniform consistency dimension-invariant L1 consistency

Cite

@article{arxiv.2607.16092,
  title  = {Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator},
  author = {Felix Gnettner and Hyemin Yeon and Piotr Kokoszka},
  journal= {arXiv preprint arXiv:2607.16092},
  year   = {2026}
}