1 results
Limit laws for large
$k$
th-nearest neighbor balls
- Part of
-
- Journal:
- Journal of Applied Probability / Volume 59 / Issue 3 / September 2022
- Published online by Cambridge University Press:
- 06 July 2022, pp. 880-894
- Print publication:
- September 2022
-
- Article
- Export citation
-
Let
$X_1,X_2, \ldots, X_n$
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.
