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On the peeling procedure applied to a Poisson point process

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Y. Davydov ,
A. Nagaev  and
A. Philippe
Y. Davydov*
Affiliation:
Université de Lille 1
A. Nagaev
Affiliation:
Nicolaus Copernicus University
A. Philippe*
Affiliation:
Université de Nantes
*
∗ Postal address: Laboratoire Paul Painlevé, Université de Lille 1 Batiment M2, 59655 Villeneuve d'Ascq Cedex, France.
∗∗ Postal address: Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex 3, France. Email address: [email protected]
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Abstract

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In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

Keywords

Control measureconvex hulllimiting shapepeelingPoisson point processstable vector

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G52: Stable processes 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 620 - 630
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

The main results of this paper were obtained together with Alexander Nagaev, with whom the first author had collaborated for more than 35 years, until Alexander's tragic death in 2005. Since then, we have gathered strength and finalised this paper, strongly feeling Alexander's absence—our memories of him will stay with us forever.

References

Davydov, Yu. and Nagaev, A. V. (2004). On the role played by extreme summands when a sum of independent and identically distributed random vectors is asymptotically α-stable. J. Appl. Prob. 41, 437–454.Google Scholar
Davydov, Yu., Molchanov, I. and Zuev, S. (2008). Strictly stable laws on convex cones. Electron. J. Prob. 13, 259–321.Google Scholar
LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Prob. 9, 624–632.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar