A Strong Law for the Largest Nearest‐Neighbour Link between Random Points
A Strong Law for the Largest Nearest‐Neighbour Link between Random Points
复制标题
随机点之间最大最近邻链接的强定律
DOI：
10.1112/s0024610799008157
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
M. Penrose
中科院分区：
文献类型：
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作者：
M. Penrose
假设X1， X2, X3，…是Rd中具有公共密度f的独立随机点，具有紧支持Ω，平滑边界∂Ω， f∣Ω连续。设Rni， k表示前n个点中从Xi到第k近邻的距离，设Mn， k = maxi≤n Rni, k。设θ表示单位球的体积。然后当n→∞,nθMn, kd / logn→马克斯(地理(minfΩ)1、2(1还是1 / d) (minf∂Ω)量1),almostsurely。
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∣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⩽n Rni, k. Let θ denote the volume of the unit ball. Then as n → ∞, nθMn,kd/logn→max((minfΩ)‐1,2(1‐1/d)(minf∂Ω)‐1),almostsurely.