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Growth Rates of Euclidean Minimal Spanning Trees with Power Weighted Edges

J. Michael Steele
The Annals of Probability
Vol. 16, No. 4 (Oct., 1988), pp. 1767-1787
Stable URL: http://www.jstor.org/stable/2243991
Page Count: 21
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Growth Rates of Euclidean Minimal Spanning Trees with Power Weighted Edges
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Abstract

Let $X_i, 1 \leq i < \infty$, 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 ψ(|Xi - Xj|). Here |Xi - Xj| denotes the Euclidean distance between Xi and Xj and ψ is a monotone function. For bounded random variables and $0 < \alpha < d$, it is proved that as n→∞ one has Mn ∼ c(α, d)n(d - α)/d ∫Rd f(x)(d-α)/d dx with probability 1, provided ψ(x) ∼ xα as x→ 0. Here f(x) is the density of the absolutely continuous part of the distribution of the {Xi}.

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