The empirical measure resulting from the nearest neighbors to a given point - \textit{the nearest neighbor measure} - is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a (local) bracketing entropy condition on the underlying class of functions (reflecting the localizing nature of the nearest neighbor algorithm). Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is equal to the conditional covariance operator (given the point of interest). This suggests the possibility of extending standard approaches - non local - replacing simply the standard empirical measure by the nearest neighbor measure while using the same way of making inference but with the nearest neighbors only instead of the full data.

翻译：由给定点的最近邻产生的经验测度——称为最近邻测度——被引入并作为核心统计量进行研究。首先，在底层函数类满足（局部）括号熵条件（反映最近邻算法的局部化性质）下，证明关联的经验过程满足一致中心极限定理。其次，在均匀熵数满足一个著名条件（常称为Vapnik-Chervonenkis条件）下，建立一致非渐近界。一致中心极限定理中高斯极限的协方差等于条件协方差算子（给定兴趣点）。这表明可能将标准非局部方法进行扩展：仅用最近邻测度替代标准经验测度，同时使用相同的推断方式，但仅基于最近邻而非全部数据。