The Annals of Applied Probability

The central limit theorem for weighted minimal spanning trees on random points

Harry Kesten and Sungchul Lee

Full-text: Open access

Abstract

Let ${X_i, 1 \leq i < \infty}$ be i.i.d. with uniform distribution on $[0, 1]^d$ and let $M(X_1, \dots, X_n; \alpha)$ be $\min {\sum_{e \epsilon T'} |e|^{\alpha}; T' \text{a spanning tree on ${X_1, \dots, X_n}$}}$. Then we show that for $\alpha > 0$, $$\frac{M(X_1, \dots, X_n; \alpha) - EM (X_1, \dots, X_n; \alpha)}{n^{(d-2 \alpha)/2d}} \to N(0, \sigma_{\alpha, d}^2)$$ in distribution for some $\sigma_{\alpha, d}^2 > 0$.

Article information

Source
Ann. Appl. Probab. Volume 6, Number 2 (1996), 495-527.

Dates
First available in Project Euclid: 18 October 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1034968141

Digital Object Identifier
doi:10.1214/aoap/1034968141

Mathematical Reviews number (MathSciNet)
MR1398055

Zentralblatt MATH identifier
0862.60008

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems

Keywords
Minimal spanning tree central limit theorem

Citation

Kesten, Harry; Lee, Sungchul. The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 (1996), no. 2, 495--527. doi:10.1214/aoap/1034968141. http://projecteuclid.org/euclid.aoap/1034968141.


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.
  • Alexander, K. S. (1994). Rates of convergence of means for distance-minimizing subadditive Euclidean functionals. Ann. Appl. Probab. 4 902-922.
  • Alexander, K. S. (1996). The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 466-494.
  • Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometric probability. Ann. Appl. Probab. 3 1033-1046.
  • Chartrand, G. and Lesniak, L. (1986). Graphs and Digraphs. Wadsworth, Belmont, CA.
  • Janson, S. (1995). The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Preprint.
  • Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7 48-50.
  • L´evy, P. (1937). Th´eorie de l'Addition des Variables Al´eatoires. Gauthier-Villars, Paris.
  • McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620-628.
  • Ramey, D. B. (1982). A non-parametric test of bimodality with applications to cluster analysis. Ph.D. dissertation, Yale Univ.
  • Redmond, C. and Yukich, J. E. (1994). Limit theorems and rates of convergence for Euclidean functionals. Ann. Appl. Probab. 4 1057-1073.
  • Steele, J. M. (1988). Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767-1787.
  • Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. vol. 81, Inst. des Mautes Etudes Scientifiques.