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

• URL: http://arxiv.org/abs/2607.16092v1
• Date: Fri, 17 Jul 2026 16:23:04 GMT
• Title: Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator
• Abstract summary: We provide a proof that the empirical spatial distribution estimator in $mathbb Rd$ and the corresponding plug-in estimator of the spatial depth are uniformly $L1$-consistent.<n>The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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