A Strong Law for the Largest Nearest-Neighbour Link between Random Points

作者: Mathew D. Penrose

DOI: 10.1112/S0024610799008157

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摘要: Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω smooth boundary ∂Ω, f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst first n points, and let Mn, = maxi[les ]nRni, k. θ volume of unit ball. Then as → ∞,formula hereIf instead lie a d-dimensional Riemannian manifold K, then nθMdn, k/ log (minKf)−1, almost surely.

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