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

arXiv: https://arxiv.org/abs/2607.16092 · HTML: https://mlsanity.com/paper/2607.16092/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 20 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [On Size-Independent Sample Complexity of ReLU Networks](https://mlsanity.com/paper/2306.01992/) (2023)
- [On data depth in infinite dimensional spaces](https://mlsanity.com/paper/1402.2775/) (2014)
- [The spatial distribution in infinite dimensional spaces and related quantiles and depths](https://mlsanity.com/paper/1402.3480/) (2014)
- [Regularized Halfspace Depth for Functional Data](https://mlsanity.com/paper/2311.07034/) (2023)

## Abstract (arXiv)

We provide a proof that the empirical spatial distribution estimator in mathbb Rd as well as the corresponding plug-in estimator of the spatial depth are uniformly L1-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.