  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 [10], 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.
