AI 中文总结
研究\(\mathbb{R}^d\)中经验空间分布估计器及相关空间深度估计器，证明其一致\(L^1\)一致，一致性速率仅依赖样本大小\(n\)，不依赖维度\(d\)等参数，源于与ChatGPT相关实验。
AI 中文摘要
我们证明了\(\mathbb{R}^d\)中的经验空间分布估计器以及空间深度的相应插件估计器是一致\(L^1\)一致的。一致性速率仅取决于样本大小\(n\)，而不取决于维度\(d\)或任何调整或正则化参数。此为罕见属性。本结果源于与ChatGPT 5.4 Pro的对话，是我们早期关于其数学推理能力实验的一部分。
英文摘要
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.
