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

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