LIMIT LAWS FOR LARGE KTH-NEAREST NEIGHBOR BALLS

Let X-1, X-2, ..., X-n be a sequence of independent random points in R-d with common Lebesgue density f. Under some conditions on f, we obtain a Poisson limit theorem, as n oo, for the number of large probability kth-nearest neighbor balls of X-1, ..., X-n. Our result generalizes Theorem 2.2 of [11 ], which refers to the special case k = 1. Our proof is completely different since it employs the Chen-Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.