Author:
Chenavier Nicolas,Henze Norbert,Otto Moritz
Abstract
AbstractLet
$X_1,X_2, \ldots, X_n$
be a sequence of independent random points in
$\mathbb{R}^d$
with common Lebesgue density f. Under some conditions on f, we obtain a Poisson limit theorem, as
$n \to \infty$
, for the number of large probability kth-nearest neighbor balls of
$X_1,\ldots, X_n$
. 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.
Publisher
Cambridge University Press (CUP)
Subject
Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability
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