#
Extremes for the Minimal Spanning Tree
on Normally Distributed Points

##
By Mathew D. Penrose.

Let *n* points be placed independently in *d*-dimensional
space according
to the standard *d*-dimensional normal distribution. Let
*M*_{n} be the
longest edge-length of the minimal spanning tree on these points;
equivalently let *M*_{n} be the infimum of those *r*
such that the union
of balls of radius *r/2* centred at the points is connected. We show that
the distribution of
*(2 *log *n)*^{1/2} M_{n} - b_{n} converges
weakly to the Gumbel (double exponential) distribution,
where *b*_{n} are
explicit constants which are asymptotic to
*(d-1) * log log *n*
as *n* becomes large.
We also show the same result holds if *M*_{n} is the longest
edge-length
for the nearest neighbour graph on the points.
Advances in Applied Probability 30, 628-639 (1998).