The limit distribution of the maximum probability nearest neighbor ball

TL;DR

This paper establishes a Poisson limit theorem and Gumbel distribution for the maximum probability of nearest neighbor balls in high-dimensional data, providing insights into extreme value behavior under mild density conditions.

Contribution

It introduces a novel limit theorem for the maximum probability nearest neighbor ball and derives a universal upper tail bound independent of the density function.

Findings

Poisson limit theorem for large probability nearest neighbor balls

Gumbel distribution for the scaled maximum probability

Density-independent upper tail bound

Abstract

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 $f$.