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

Gupta, Bhupender; Iyer, Srikanth K.

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

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Description

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 \[\lim_{n \rar \infty} \frac{\sqrt{\log n} d_n}{\log \log n} = \frac{d}{\sqrt{2}}, \qquad d \geq 2, {a.s.} \]
10 pages