The law of the iterated logarithm for the total length of the nearest neighbor graph
type: Item
instance of: scholarly article
title: The law of the iterated logarithm for the total length of the nearest neighbor graph
zbMATH Open document ID: 1053.60026
author: Sungchul Lee
author: Young-Hoon Kim
author: Zhengyan Lin
published in: Journal of Theoretical Probability
publication date: 2004-08-06T00:00:00Z
review text: Consider a Poisson point process \({\mathcal{P}}_n\) on \([-n,n]^d\) with intensity \(1\). For every Poisson point \(x=(x_1,\ldots, x_d)\in {\mathcal{P}}_n\) let \(y\in {\mathcal{P}}_n \setminus \{x\}\) denote the nearest neighbor to \(x\) (in Euclidean distance). Set \(L_n=\sum_{(x,y)\in{\mathcal{E}}_n} | x-y|\), where \({\mathcal{E}}_n\) denotes the collection of pairs \((x,y)\) of Poisson points and their respective nearest neighbors. The authors' main result proves that \[ \mathop{\limsup}\limits_{n\rightarrow\infty} {{L_n-EL_n}\over{\sqrt{2\sigma^2_n \log\log \sigma^2_n}}}=1\quad\text{a.s.,} \] where \(\sigma^2_n=\text{Var} (L_n)\). Since, by a standard argument, \(\sigma^2_n \sim n^d \sigma^2\) with some positive constant \(\sigma^2\), the above LIL can be reformulated as \[ \limsup_{n\rightarrow\infty} {{L_n-EL_n}\over{\sqrt{2\sigma^2 n^d \log\log n^d}}}=1\quad\text{a.s.} \]
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 2083468
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: Poisson point process
zbMATH Keywords: nearest neighbor graph
zbMATH Keywords: geometric probability
zbMATH Keywords: normal approximation
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
DOI: 10.1023/B:JOTP.0000020483.81884.A2
description: scientific article; zbMATH DE number 2083468