Nearest neighbor process: weak convergence and non-asymptotic bound

An empirical measure that results from the nearest neighbors to a given point - the nearest neighbor measure - is introduced and studied as a central statistical quantity. First, the resulting 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 nearest neighbor algorithm). Second a uniform non-asymptotic bound is established under a well-known condition, often refereed to as Vapnik-Chervonenkis, on the uniform entropy numbers.