A Strong Law for the Largest Nearest-Neighbor Link on Normally Distributed Points

TL;DR

This paper establishes a strong law describing the asymptotic behavior of the longest nearest-neighbor link in a graph formed by points sampled from a standard normal distribution in high-dimensional space.

Contribution

It provides a precise almost sure limit for the longest nearest-neighbor edge length in a high-dimensional normal sample, extending understanding of geometric properties of random point sets.

Findings

The normalized longest nearest-neighbor link converges almost surely.

The limit involves the dimension and logarithmic factors.

Results hold for dimensions greater than or equal to 2.

Abstract

Let $n$ points be placed independently in $d-$dimensional space according to the standard $d-$dimensional normal distribution. Let $d_n$ be the longest edge length for the nearest neighbor graph on these points. We show that \[\lim_{n \rar \infty} \frac{\sqrt{\log n} d_n}{\log \log n} = \frac{d}{\sqrt{2}}, \qquad d \geq 2, {a.s.} \]