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

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 $$\underset{n\text{rar}\mathrm{\infty }}{\lim }\frac{\sqrt{\log n}{d}_n}{\log \log n}=\frac{d}{\sqrt{2}},d\ge 2,a\mathrm{.}s\mathrm{.}$$