提出新框架，解决复杂分布族的均值统一估计问题。
Uniform Convergence Beyond Glivenko-Cantelli
- 用任意估计器替代经验均值，定义均值可统一估计性
- 均值向量可分是充分条件，但非必要条件
- 证明可数并集仍保持可估计性，解决此前猜想
我们刻画了在{0,1}^N上分布族实现均值统一估计的条件。以往工作聚焦于使用经验均值估计的统一收敛性，即P-Glivenko-Cantelli原理。本文突破该框架，不再局限于经验均值，提出统一均值可估计性（UME-learnability），衡量任意估计器能否实现均值的统一估计。研究对象为分布族的均值向量空间——每个分布对应一个记录各坐标期望值的向量。我们证明：均值向量可分是UME-learnability的充分条件；但通过构造反例，说明可分性非必要——存在非可分均值向量的分布族，仍可通过根本不同的技术实现UME-学习。最后，我们证明：任意可数个UME-可学习集合的并集仍是UME-可学习的，解决了Cohen等（2025）提出的猜想。
原文摘要 · Abstract (English)
We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ admit uniform estimation of their mean. Prior work from Vapnik and Chervonenkis (1971) has focused on uniform convergence using the empirical mean estimator, leading to the principle known as $P-$ Glivenko-Cantelli. We extend this framework by moving beyond the empirical mean estimator and introducing Uniform Mean Estimability, also called UME-learnability, which captures when a collection permits uniform mean estimation by any arbitrary estimator. We work on the space created by the mean vectors of the collection of distributions. For each distribution, the mean vector records the expected value in each coordinate. We show that separability of the mean vectors is a sufficient condition for UME-learnability. However, we show that separability of the mean vectors is not necessary for UME-learnability by constructing a collection of distributions whose mean vectors are non-separable yet UME-learnable using techniques fundamentally different from those used in our separability-based analysis. Finally, we establish that countable unions of UME-learnable collections are also UME-learnable, solving the conjecture posed in Cohen et al. (2025).
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