The Annals of Probability

Extremes on trees

Tailen Hsing and Holger Rootzén

Full-text: Open access


This paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on X1,…,XNn where X1,X2,…  are i.i.d. in ℜ2 with distribution F and Nn is independent of the Xi and satisfies Nn/np1. A new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which F has unbounded support. The general results are applied to a number of special cases, including elliptically contoured distributions, distributions with independent Weibull-like margins and distributions with parallel level curves.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 413-444.

First available in Project Euclid: 11 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Extreme values minimal spanning tree nearest neighbor graph


Hsing, Tailen; Rootzén, Holger. Extremes on trees. Ann. Probab. 33 (2005), no. 1, 413--444. doi:10.1214/009117904000001008.

Export citation


  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Erdelyi, A. (1956). Asymptotic Expansions. Dover, New York.
  • Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
  • Kallenberg, O. (1983). Random Measures. Academic Press, New York.
  • Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495--527.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab. 7 996--1020.
  • Penrose, M. D. (1997). The longest edge of the minimal spanning tree. Ann. Appl. Probab. 7 340--361.
  • Penrose, M. D. (1998). Extremes for the minimal spanning tree on normally distributed points. Adv. in Appl. Probab. 30 628--639.
  • Penrose, M. (2000). Central limit theorems for $k$-nearest neighbour distances. Stochastic Process. Appl. 85 295--320.