  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 $f$.
