math.PR

# Title: Limit laws for large kth-nearest neighbor balls

Abstract: Let $X_1,\ldots,X_n$ 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 $k$th-nearest neighbor balls of $X_1,\ldots,X_n$. Our result generalizes Theorem 2. of , 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.
 Comments: 11 pages Subjects: Probability (math.PR) MSC classes: 60F05 (Primary), 60D05 (Secondary) Cite as: arXiv:2105.00038 [math.PR] (or arXiv:2105.00038v1 [math.PR] for this version)

## Submission history

From: Norbert Henze [view email]
[v1] Fri, 30 Apr 2021 18:34:05 GMT (16kb)

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