Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator
Felix Gnettner, Hyemin Yeon, Piotr Kokoszka
math.ST math.PR stat.ML
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.
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Classified with taxonomy v2 on Sat, 5 Sept 2026.