The limit distribution of the maximum probability nearest neighbor ball.

Let $X_1, ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large probability nearest neighbor balls. Denoting by $P_n$ the maximum probability measure of nearest neighbor balls, this limit theorem implies a Gumbel extreme value distribution for $nP_n - ln n$ as $n to infty$. Moreover, we derive a tight upper bound on the upper tail of the distribution of $nP_n - ln n$, which does not depend on $f$.