[Paper Review] A Strong Law for the Largest Nearest-Neighbor Link on Normally Distributed Points

Bhupender Gupta, Srikanth K. Iyer|ArXiv.org|Apr 27, 2006

Point processes and geometric inequalities3 references3 citations

TL;DR

This paper establishes a strong law of large numbers for the largest nearest-neighbor distance $ d_n $ in a nearest-neighbor graph formed by $ n $ i.i.d. standard normal points in $ \mathbb{R}^d $, showing that $ \frac{\sqrt{\log n}\, d_n}{\log \log n} \to \frac{d}{\sqrt{2}} $ almost surely as $ n \to \infty $, for $ d \geq 2 $. The result is derived via Poisson process coupling, spatial concentration bounds, and Borel-Cantelli arguments on annular regions of increasing radius.

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 \[\lim_{n ar \infty} \frac{\sqrt{\log n} d_n}{\log \log n} = \frac{d}{\sqrt{2}}, \qquad d \geq 2, {a.s.} \]

Motivation & Objective

• To establish a strong law for the largest nearest-neighbor distance in a nearest-neighbor graph formed by $ n $ i.i.d. standard normal points in $ \mathbb{R}^d $, $ d \geq 2 $.
• To extend strong law results beyond compactly supported densities to the unbounded support case of the multivariate normal distribution.
• To analyze the asymptotic behavior of the longest edge in the nearest-neighbor graph via spatial concentration and Poisson process coupling.
• To resolve the scaling of the largest nearest-neighbor distance in high-dimensional normal point processes.

Proposed method

• Couple the i.i.d. point process $ \mathcal{X}_n $ with an inhomogeneous Poisson process $ \mathcal{P}_n $ of intensity $ n\phi(\cdot) $, ensuring $ \mathcal{P}_n^{-} \subset \mathcal{X}_n \subset \mathcal{P}_n^{+} $ for large $ n $.
• Define annular regions $ A_n = B(0, R_n(c)) \setminus B(0, R_n'(-2)) $, where $ R_n(c) $ captures the typical radial extent of $ n $ normal points.
• Use Lemma 2.1 to derive asymptotic approximations for the probability that a ball of radius $ r_n $ centered at a point of radius $ \rho_n $ contains no points.
• Apply the Borel-Cantelli lemma to show that the events $ \{ \mathcal{X}_n \subset B(0, R_n(c)) \} $ and $ \{ \mathcal{X}_n \cap B^c(0, R_n(c)) \neq \emptyset \} $ occur only finitely often for $ c > 2 $ and $ c < 0 $, respectively.
• Construct disjoint balls $ B(x_i^n, r_n(u)) $ within $ A_n $ and define events $ E_n(x_i^n) $ indicating exactly one point of $ \mathcal{X}_n $ in a small ball and none in a larger one.
• Bound the probability of $ \bigcup_{i=1}^{\sigma_n} E_n(x_i^n) $ and show it is summable, implying that $ d_n \geq r_n(t) $ infinitely often a.s. for $ t < \frac{2d + c - 2}{2\sqrt{2}} $.

Experimental results

Research questions

• RQ1 What is the almost sure asymptotic scaling of the largest nearest-neighbor distance in a nearest-neighbor graph of $ n $ i.i.d. standard normal points in $ \mathbb{R}^d $, $ d \geq 2 $?
• RQ2 How does the radial concentration of normal points affect the extremal behavior of nearest-neighbor distances?
• RQ3 Can strong law results be established for the largest nearest-neighbor distance when the underlying density has unbounded support, such as the normal distribution?
• RQ4 What is the role of Poisson process coupling and spatial annular decomposition in deriving such strong laws for geometric graph parameters?
• RQ5 How do the asymptotics of the normal density's tail behavior influence the scaling of the longest edge in the nearest-neighbor graph?

Key findings

• The largest nearest-neighbor distance $ d_n $ satisfies $ \lim_{n \to \infty} \frac{\sqrt{16 \log n} \, d_n}{\log \log n} = \frac{d}{\sqrt{2}} $ almost surely for $ d \geq 2 $, with $ \log \log n $ denoted as $ \log_2 n $ in the paper.
• The radial extent of $ n $ standard normal points concentrates around $ R_n(c) = \sqrt{2\log n + (c + d - 2)\log \log n + 2\log A_d} $, with $ A_d = (2\pi)^{-d/2}/d $.
• For $ c > 2 $, all $ \mathcal{X}_n $ points lie within $ B(0, R_n(c)) $ eventually almost surely, and for $ c < 0 $, at least one point lies outside $ B(0, R_n(c)) $ infinitely often a.s.
• The probability that a point in annular region $ A_n $ has no other point within distance $ r_n(u) $ is bounded below by a term decaying slower than any polynomial, enabling summability via Borel-Cantelli.
• The construction of disjoint balls $ B(x_i^n, r_n(u)) $ in $ A_n $ ensures that the number of such balls $ \sigma_n $ grows as $ \left( \frac{\log n}{\log \log n} \right)^{d-1} $, capturing the spatial density of points.
• The proof establishes that $ d_n \geq r_n(t) $ infinitely often a.s. for $ t < \frac{2d + c - 2}{2\sqrt{2}} $, and by optimizing $ c $, the limit $ \frac{d}{\sqrt{2}} $ is achieved.