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The empirical measure resulting from the nearest neighbors to a given point -\textit{the nearest neighbor measure} - is introduced and studied as a centralstatistical quantity. First, the associated empirical process is shown tosatisfy a uniform central limit theorem under a (local) bracketing entropycondition on the underlying class of functions (reflecting the localizingnature of the nearest neighbor algorithm). Second a uniform non-asymptoticbound is established under a well-known condition, often referred to asVapnik-Chervonenkis, on the uniform entropy numbers. The covariance of theGaussian limit obtained in the uniform central limit theorem is equal to theconditional covariance operator (given the point of interest). This suggeststhe possibility of extending standard approaches - non local - replacing simplythe standard empirical measure by the nearest neighbor measure while using thesame way of making inference but with the nearest neighbors only instead of thefull data.

Nearest neighbor process: weak convergence and non-asymptotic bound