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

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

2006-04-27

arxiv.org/abs/math/0604585v1

Mathematics

Probability

10 pages

Scientific paper

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

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