A Strong Law for the Largest Nearest-Neighbor Link on Normally Distributed Points

2006/04/27 by Bhupender Gupta, Gupta, Bhupender, Srikanth K. Iyer +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.math/0604585

10 pages

arxiv created 2006/04/27 · openalex publication_date 2006/04/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n points be placed independently in d-dimensional space according to the standard d-dimensional normal distribution. Let d_n be the longest edge length for the nearest neighbor graph on these points. We show that lim_{n \rar ∞} (√(log n) d_n)/(log log n) = (d)/(√(2)), d ≥ 2, a.s.

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