The Annals of Applied Probability

The central limit theorem for Euclidean minimal spanning trees I I

Sungchul Lee

Full-text: Open access

Abstract

Let ${X_i: i \geq 1}$ be i.i.d. with uniform distribution $[- 1/2, 1/2]^d, d \geq 2$, and let $T_n$ be a minimal spanning tree on ${X_1, \dots, X_n}$. For each strictly positive integer $\alpha$, let $N({X_1, \dots, X_n}; \alpha)$ be the number of vertices of degree $\alpha$ in $T_n$. Then, for each $\alpha$ such that $P(N({X_1, \dots, X_{\alpha+1}}; \alpha) = 1) > 0$, we prove a central limit theorem for $N({X_1, \dots, X_n}; \alpha)$.

Article information

Source
Ann. Appl. Probab., Volume 7, Number 4 (1997), 996-1020.

Dates
First available in Project Euclid: 29 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1043862422

Digital Object Identifier
doi:10.1214/aoap/1043862422

Mathematical Reviews number (MathSciNet)
MR1484795

Zentralblatt MATH identifier
0892.60034

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C05: Trees 90C27: Combinatorial optimization

Keywords
Minimal spanning tree central limit theorem continuum percolation

Citation

Lee, Sungchul. The central limit theorem for Euclidean minimal spanning trees I I. Ann. Appl. Probab. 7 (1997), no. 4, 996--1020. doi:10.1214/aoap/1043862422. https://projecteuclid.org/euclid.aoap/1043862422


Export citation

References

  • ALDOUS, D. and STEELE, J. M. 1992. Asy mptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247 258. Z.
  • ALEXANDER, K. S. 1995. Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Comm. Math. Phy s. 168 39 55. Z.
  • ALEXANDER, K. S. 1996. The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 466 494. Z.
  • BENTLEY, J. L. and FRIEDMAN, J. H. 1978. Fast algorithms for constructing minimal spanning trees in coordinate spaces. IEEE Comput. 27 97 105. Z.
  • CHANG, C. L., CHANG, S. K., KANG, A. N. C. and LEE, R. C. T. 1977. Storage reduction through minimal spanning trees and spanning forests. IEEE Comput. 26 425 434. Z.
  • CHIN, F. Y. and HOUCK, D. J. 1978. Algorithms for updating minimal spanning trees. J. Comput. Sy stem Sci. 16 333 344. Z.
  • DUSSERT, C., LLEBARIA, A., MARTY, F., PALMARI, J., RASIGNI, G. and RASIGNI, M. 1987. Minimal spanning tree analysis of biological structures. J. Theoret. Biol. 125 317 323. Z.
  • EDDY, W. F., SHEPP, L. A. and STEELE, J. M. 1987. On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab. 24 809 826. Z.
  • FRIEDMAN, J. H. and RAFSKY, L. C. 1979. Multivariate generalizations of the Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7 697 717. Z.
  • FRIEDMAN, J. H. and RAFSKY, L. C. 1983. Graph-theoretic measures of multivariate association and prediction. Ann. Statist. 11 377 391. Z.
  • HALL, P. and HEy DE, C. C. 1980. Martingale Limit Theory and Its Application. Academic Press, New York. Z.
  • JUNG, H. A. 1974. Determination of minimal paths and spanning trees in graphs. Computing 13 249. Z.
  • KATAJAINEN, J. 1983. On the worst case of a minimal spanning tree algorithm for Euclidean space. BIT 23 2 8.
  • KESTEN, H. and LEE, S. 1996. The central limit theorem for weighted minimal spanning trees on random point. Ann. Appl. Probab. 6 495 527. Z.
  • KRUSKAL, J. B. 1956. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7 48 50. Z.
  • LEVY, P. 1937. Theorie de l' Addition des Variables Aleatoires. Gauthier-Villars, Paris. ´ ´ ´ Z.
  • MALLION, R. B. 1975. Number of spanning trees in a molecular graph. Chem. Phy s. Lett. 36 170 174. Z.
  • MCLEISH, D. L. 1974. Dependent central limit theorems and invariance principles. Ann. Probab. 2 620 628. Z.
  • PENNY, D. 1980. Techniques for the verification of minimal phy logenetic trees illustrated with 10 mammalian hemoglobin sequences. Biochem. J. 187 65 74. Z.
  • PENROSE, M. D. 1996. The random minimal spanning tree in high dimensions. Ann. Probab. 6 528 544. Z.
  • REDMOND, C. and YUKICH, J. E. 1994. Limit theorems and rates of convergence for Euclidean functionals. Ann. Appl. Probab. 4 1057 1073. Z.
  • RHEE, W. T. and TALAGRAND, M. 1989. A sharp deviation inequality for the stochastic traveling salesman problem. Ann. Probab. 17 1 8. Z.
  • ROHLF, F. J. 1975. Generalization of the gap test for the detection of multivariate outliers. Biometrics 31 93 101. Z.
  • ROMANE, F. 1977. Possible use of minimum spanning tree in phy toecology. Vegetation 33 99 106. Z.
  • ROSEN, K. H. 1995. Discrete Mathematics and Its Applications, 3rd ed. McGraw-Hill, New York.Z.
  • STEELE, J. M. 1988. Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767 1787. Z.
  • WHITNEY, V. K. M. 1972. Minimal spanning tree. Comm. ACM 15 273. Z.
  • WU, F. Y. 1977. Number of spanning trees on a lattice. J. Phy s. A 10 L113 L115.