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The central limit theorem for Euclidean minimal spanning trees II

Part of: Geometric probability and stochastic geometry Limit theorems Graph theory Special processes

Published online by Cambridge University Press:  01 July 2016

Sungchul Lee
Sungchul Lee*
Affiliation:
Yonsei University
*
∗ Postal address: Department of Mathematics, Yonsei University, Seoul 120-749, Korea. Email address: sungchul@ bubble.yonsei.ac.kr
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Abstract

Let Xi : i ≥ 1 be i.i.d. points in ℝd, d ≥ 2, and let Tn be a minimal spanning tree on X1,…,Xn. Let L(X1,…,Xn) be the length of Tn and for each strictly positive integer α let N(X1,…,Xn;α) be the number of vertices of degree α in Tn. If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X1,…,Xn) and N(X1,…,Xn;α). We also study the rate of convergence for EL(X1,…,Xn).

Keywords

Minimal spanning treecentral limit theoremcontinuum percolation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 05C05: Trees 90C27: Combinatorial optimization
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 969 - 984
Copyright
Copyright © Applied Probability Trust 1999 

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References

Alexander, K. S. (1994). Rates of convergence of means for distance-minimizing subadditive Euclidean functionals. Ann. Appl. Prob. 4, 902–922.CrossRefGoogle Scholar
Alexander, K. S. (1995). Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Commun. Math. Phys. 168, 39–55.CrossRefGoogle Scholar
Friedman, J. H. and Rafsky, L. C. (1979). Multivariate generalizations of the Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7, 697–717.CrossRefGoogle Scholar
Friedman, J. H. and Rafsky, L. C. (1983). Graph-theoretic measures of multivariate association and prediction. Ann. Statist. 11, 377–391.CrossRefGoogle Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Jaillet, P. (1993). Rate of convergence for the Euclidean minimum spanning tree law. Operat. Res. Lett. 14, 73–78.CrossRefGoogle Scholar
Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random point. Ann. Appl. Prob. 6, 495–527.CrossRefGoogle Scholar
Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees I. Ann. Appl. Prob. 7, 996–1020.CrossRefGoogle Scholar
Lévy, P., (1937). Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris.Google Scholar
McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Prob. 2, 620–628.CrossRefGoogle Scholar
Prim, R. C. (1957). Shortest connecting networks and some generalizations. Bell Syst. Tech. J. 36, 1389–1401.CrossRefGoogle Scholar
Rohlf, F. J. (1975). Generalization of the gap test for the detection of multivariate outliers. Biometrics 31, 93–101.CrossRefGoogle Scholar
Steele, J. M. (1988). Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Prob. 16, 1767–1787.CrossRefGoogle Scholar
Steele, J. M., Shepp, L. A. and Eddy, W. F. (1987). On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Prob. 24, 809–826.CrossRefGoogle Scholar