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

Published online by Cambridge University Press:  01 December 1999

MATHEW D. PENROSE
Affiliation:
Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE; [email protected]
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Abstract

Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω with smooth boundary ∂Ω, with f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst the first n points, and let Mn, k = maxi[les ]nRni, k. Let θ denote the volume of the unit ball. Then as n → ∞,

formula here

If instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/ log n → (minKf)−1, almost surely.

Type
Notes and Papers
Copyright
The London Mathematical Society 1999

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