Abstract
It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk.
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Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. National Bureau of Standards No. 55, Washington D.C., 1970
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Billingsley, P.: Probability and measure. New York: Wiley 1979
Buchta, C.: Stochastische Approximation konvexer Polygone. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 283–304 (1984)
Buchta, C.: Zufällige Polyeder, eine Übersicht. In: Hlawka, E. (ed.) Zahlentheoretische Analysis. Wiener Seminarberichte 1980–82. (Lect. Notes Math., vol. 1114, pp. 1–13) Berlin Heidelberg New York: Springer 1985
Eddy, W.F., Gale, J.D.: The convex hull of a spherically symmetric sample. Adv. Appl. Prob. 13, 751–763 (1981)
Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–343 (1965)
Geffroy, J.: Contribution à la théorie des valeurs extrémes. Publ. Inst. Stat. Univ. Paris VIII, 123–185 (1959)
Geffroy, J.: Localisation asymptotique du polyèdre d'appui d'un échantillon Laplacien à k dimensions. Publ. Inst. Stat. Univ. Paris X, 212–228 (1961)
Ibragimow, I.A., Linnik, Y.V.: Independent and stationary sequences of random variables. Groningen: Wolters Noordhoff 1971
Kallenberg, O.: Random measures. Berlin: Akademie Verlag 1983
Raynaud, H.: Sur l'enveloppe convexe des nuages de points aléatoires dans ℝn. J. Appl. Probab. 7, 35–48 (1970)
Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 75–84 (1963)
Schneider, R.: Random approximation of convex sets. Preprint Mathematical Institute, Albert-Ludwigs University, Freiburg im Breisgau, FRG (1987)
Stroock, D.W.: Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 209–244 (1975)
Vervaat, W.: Upper bounds for the distance in total variation between the binomial or negative binomial and the Poisson distribution. Stat. Neerl. 23, 79–86 (1969)
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Groeneboom, P. Limit theorems for convex hulls. Probab. Th. Rel. Fields 79, 327–368 (1988). https://doi.org/10.1007/BF00342231
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DOI: https://doi.org/10.1007/BF00342231