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A STRONG LAW FOR THE LONGEST EDGE OF THE MINIMAL SPANNING TREE

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By Mathew D. Penrose.

Suppose X_{1},X_{2},X_{3},...
are independent random points in *d*-dimensional
space with common density *f*, having compact
connected support *A* with smooth boundary.
Suppose the restriction of
*f* to *A* is continuous.
Let *M*_{n}
be the smallest *r* such that the union of balls of diameter
*r* centred at the first *n* points is connected.
We
derive an almost sure limit for * n
(M*_{n})^{d}/ log * n *.
Annals of Probability 27, 290-298 (1999).