Statistics Papers
Document Type
Journal Article
Date of this Version
1988
Publication Source
The Annals of Probability
Volume
16
Issue
4
Start Page
1767
Last Page
1787
DOI
10.1214/aop/1176991596
Abstract
Let Xi, 1 ≤ i < ∞, denote independent random variables with values in Rd, d ≥ 2, and let Mn denote the cost of a minimal spanning tree of a complete graph with vertex set {X1, X2, . . . , Xn}, where the cost of an edge (Xi, Xj) is given by ⋺(|Xi - Xj|). Here |Xi - Xj| denotes the Euclidean distance between Xi and Xj and ⋺ is a monotone function. For bounded random variables and 0 < a < d, it is proved that as n → ∞ one has Mn ~ c(a, d)n(d-a)/d∫Rdf(x)(d-a)/d dx with probability 1, provided ⋺(x) ~ xa as x → 0. Here f(x) is the density of the absolutely continuous part of the distribution of the {Xi}.
Copyright/Permission Statement
The original and published work is available at: https://projecteuclid.org/euclid.aop/1176991596#abstract
Keywords
minimal spanning trees, subadditive processes
Recommended Citation
Steele, J. M. (1988). Growth Rates of Euclidean Minimal Spanning Trees With Power Weighted Edges. The Annals of Probability, 16 (4), 1767-1787. http://dx.doi.org/10.1214/aop/1176991596
Date Posted: 27 November 2017
This document has been peer reviewed.