The limit distribution of the maximum probability nearest neighbor ball

Let $X_1,\dots ,X_n$ 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 $n{P}_n-\ln n$ as $n\to \mathrm{\infty }$. Moreover, we derive a tight upper bound on the upper tail of the distribution of $n{P}_n-\ln n$, which does not depend on $f$.