Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:38:39.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Focusing of the scan statistic and geometric clique number

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

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of Durham
*
∗ Postal address: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK. Email address: mathew.penrose@durham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

Keywords

Extremesrandom graphasymptotically quasi-deterministicChen-Stein

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60D05: Geometric probability and stochastic geometry 05C80: Random graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 739 - 753
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alm, S. E. (1997). On the distributions of scan statistics of a two-dimensional Poisson process. Adv. Appl. Prob. 29, 1–18.Google Scholar
[2] Anderson, C. W., Coles, S. G. and Hüsler, J. (1997). Maxima of Poisson-like variables and related triangular arrays. Ann. Appl. Prob. 7, 953–971.Google Scholar
[3] Appel, M. J. B. and Russo, R. (1997). The maximum vertex degree of a graph on uniform points in [0,1]d . Adv. Appl. Prob. 29, 567–581.Google Scholar
[4] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25.Google Scholar
[5] Auer, P. and Hornik, K. (1994). On the number of points of a homogeneous Poisson process. J. Multivariate Anal. 48, 115–156.Google Scholar
[6] Auer, P., Hornik, K. and Révész, P. (1991). Some limit theorems for homogeneous Poisson processes. Statist. Prob. Lett. 12, 91–96.Google Scholar
[7] Barbour, A. D. and Månsson, M. (2000). Compound Poisson approximation and the clustering of random points. Adv. Appl. Prob. 32, 19–38.Google Scholar
[8] Bollobás, B., (1985). Random Graphs. Academic Press, London.Google Scholar
[9] Cressie, N. (1991). Statistics for Spatial Data. John Wiley, New York.Google Scholar
[10] Glaz, J. and Balakrishnan, N. (eds) (1999). Scan Statistics and Applications. Birkhäuser, Boston.Google Scholar
[11] Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.Google Scholar
[12] Godehardt, E. (1990). Graphs as Structural Models, 2nd edn. Vieweg, Braunschweig.Google Scholar
[13] Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
[14] McDiarmid, C. (2001). Random channel assignment in the plane. To appear in Random Structures Algorithms.Google Scholar
[15] Månsson, M. (1999). Poisson approximation in connection with clustering of random points. Ann. Appl. Prob. 9, 465–492.Google Scholar
[16] Matula, D. W. (1970). On the complete subgraph of a random graph. In Proc. 2nd Chapel Hill Conf. Combinatorial Math. Appl., University of North Carolina, Chapel Hill, pp. 356–369.Google Scholar
[17] Penrose, M. D. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Prob. 7, 340–361.Google Scholar
[18] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press (in press).Google Scholar
[19] Rudin, W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar