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On the peeling procedure applied to a Poisson point process

  • Y. Davydov (a1), A. Nagaev (a2) and A. Philippe (a3)

Abstract

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.

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Copyright

Corresponding author

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: anne.philippe@univ-nantes.fr

Footnotes

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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.

Footnotes

References

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[1] 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, 437454.
[2] Davydov, Yu., Molchanov, I. and Zuev, S. (2008). Strictly stable laws on convex cones. Electron. J. Prob. 13, 259321.
[3] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Prob. 9, 624632.
[4] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
[5] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.

Keywords

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On the peeling procedure applied to a Poisson point process

  • Y. Davydov (a1), A. Nagaev (a2) and A. Philippe (a3)

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