Central limit theorems for \(k\)-nearest neighbour distances
type: Item
instance of: scholarly article
title: Central limit theorems for \(k\)-nearest neighbour distances
zbMATH Open document ID: 0997.60014
DOI: 10.1016/S0304-4149(99)00080-0
author: Mathew D. Penrose
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: Let \(X_1,X_2,\dots \) be i.i.d. \(d\)-dimensional random vectors with a density function \(f.\) Let \(R_{i,k,n}\) denote the distance from \(X_i\) to its \(k\)th nearest neighbour in the set \(\{X_1,\dots , X_n\}, n>k\), and similarly, let \(R_{i,k, N_n}\) be the distance from \(X_i\) to its \(k\)th nearest neighbour in \(\{X_1,\dots , X_{N_n} \},\) where for each \(n =1, 2,\dots\), \(\;N_n \) is a Poisson process with mean \(n\). Weak convergence of centered and scaled empirical distributions of \(\{f(X_i)R^d_{i, k_n, n}, 1\leq i\leq n\}\), respectively of \(\{f(X_i)R^d_{i, k_n, N_n}, 1\leq i\leq N_n\}\) is studied as \(n\to \infty \) and \(k_n = o(n^{2/(2+d)})\).
reviewed by: Zuzana Prášková
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 1792505
zbMATH Keywords: nearest neighbours
zbMATH Keywords: empirical processes
zbMATH Keywords: Gaussian process
zbMATH Keywords: dependency graph
MaRDI profile type: Publication
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Wikidata QID: Q127644718
description: scientific article; zbMATH DE number 1792505