We study the limiting behavior of sums of functions of nearest neighbor distances for an $m$ dimensional sample. We establish a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances. As a consequence we obtain the asymptotic behavior of a practicable goodness of fit test based on nearest neighbor distances.
"Sums of Functions of Nearest Neighbor Distances, Moment Bounds, Limit Theorems and a Goodness of Fit Test." Ann. Probab. 11 (1) 185 - 214, February, 1983. https://doi.org/10.1214/aop/1176993668