Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T17:38:44.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Rooted edges of a minimal directed spanning tree on random points

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

Published online by Cambridge University Press:  01 July 2016

Z. D. Bai ,
Sungchul Lee  and
Mathew D. Penrose
Z. D. Bai*
Affiliation:
Northeast Normal University and National University of Singapore
Sungchul Lee*
Affiliation:
Yonsei University
Mathew D. Penrose*
Affiliation:
University of Bath
*
∗ Postal address: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P. R. China.
∗∗ Postal address: Department of Mathematics, Yonsei University, Seoul 120-749, Korea. Email address: [email protected]
∗∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For n independent, identically distributed uniform points in [0, 1]d, d ≥ 2, let Ln be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of the mean and the variance of Ln, and show that Ln satisfies a central limit theorem, unlike in the case d = 2.

Keywords

Minimal spanning treemultivariate extremecentral limit theorem

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 1 - 30
Copyright
Copyright © Applied Probability Trust 2006 

References

Bai, Z.-D., Chao, C.-C., Hwang, H.-K. and Liang, W.-Q. (1998). On the variance of the number of maxima in random vectors and its applications. Ann. Appl. Prob. 8, 886–895.CrossRefGoogle Scholar
Bai, Z.-D., Devroye, L., Hwang, H.-K. and Tsai, T.-H. (2005). Maxima in hypercubes. Random Structures Algorithms 27, 290–309.CrossRefGoogle Scholar
Bai, Z.-D. et al (2001). Limit theorems for the number of maxima in random samples from planar regions. Electron. J. Prob. 6, 41pp.CrossRefGoogle Scholar
Barbour, A. D. and Xia, A. (2001). The number of two-dimensional maxima. Adv. Appl. Prob. 33, 727–750.CrossRefGoogle Scholar
Baryshnikov, Y. (2000). Supporting points processes and some of their applications. Prob. Theory Relat. Fields 117, 163–182.CrossRefGoogle Scholar
Bhatt, A. and Roy, R. (2004). On a random directed spanning tree. Adv. Appl. Prob. 36, 19–42.CrossRefGoogle Scholar
Bräker, H. and Hsing, T. (1998). On the area and perimeter of a random convex hull in a bounded convex set. Prob. Theory Relat. Fields 111, 517–550.CrossRefGoogle Scholar
Groeneboom, P. (1988). Limit theorems for convex hulls. Prob. Theory Relat. Fields 79, 329–368.CrossRefGoogle Scholar
Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.CrossRefGoogle Scholar
Hsing, T. (1994). On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Prob. 4, 478–493.CrossRefGoogle Scholar
Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs. John Wiley, New York.CrossRefGoogle Scholar
Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Prob. 6, 495–527.CrossRefGoogle Scholar
Penrose, M. D. (2003). Random Geometric Graphs (Oxford Studies Prob. 6). Oxford University Press.Google Scholar
Penrose, M. D. and Wade, A. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. Appl. Prob. 36, 691–714.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.Google Scholar
Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems (Lecture Notes Math. 1675). Springer, Berlin.Google Scholar