Nearest neighbor process: weak convergence and non-asymptotic bound

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.