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A STRONG LAW FOR THE LARGEST NEAREST-NEIGHBOUR LINK BETWEEN RANDOM POINTS

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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
support *A* with smooth boundary.
Suppose the restriction of
*f* to *A* is continuous.
Let *R*_{i,k,n}
denote the distance from X_{i} to its *k*-th nearest
neighbour amongst the first *n* points, and let
*M*_{n,k} = max_{i <= n }
R_{i,k,n}. We
derive an almost sure limit for * n
(M*_{n,k})^{d}/ log * n *.
We give an analogous result for the case where the points lie in a compact
smooth *d*-dimensional Riemannian manifold.
Journal of the London Mathematical Society (2),
60, 951-960 (1999).