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)/dRdf(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

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Date Posted: 27 November 2017

This document has been peer reviewed.