Central limit theorems for random polytopes

Probability Theory and Related Fields

Volumn 133, Issue 4, 2005, Pages 483-507

  Reitzner, Matthias ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Approximation of convex bodies; CLT; Dependency graph; Random polytopes

Indexed keywords

EID: 27844503003     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-005-0441-8     Document Type: Article

References (31)

1. Alagar, V.S.: On the distribution of a random triangle. J. Appl. Probability 14, 284-297 (1977)
2. Avram, F., Bertsimas, D.: On central limit theorems in geometrical probability. Ann. Appl. Probab. 3, 1033-1046 (1993)
3. Baldi, P., Rinott, Y.: On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17, 1646-1650 (1989)
4. Bárány, I.: Random polytopes in smooth convex bodies. Mathematika 39, 81-92 (1992)
5. Bárány, I.: Sylvester's question: the probability that n points are in convex position. Ann. Probab. 27, 2020-2034 (1999)
6. Bárány, I.: A note on Sylvester's four-point problem. Studia Sci. Math. Hungar. 38, 73-77 (2001)
7. Bárány, I., Buchta, C.: Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann. 297, 467-497 (1993)
8. Bárány, I., Rote, G., Steiger, W., Zhang, C.-H.: A central limit theorem for convex chains in the square. Discrete Comput. Geom. 23, 35-50 (2000)
9. Buchta, C.: An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33, 125-142 (2005)
10. Cabo, A.J., Groeneboom, P.: Limit theorems for functionals of convex hulls. Probab. Theory Relat. Fields 100, 31-55 (1994)
11. Efron, B.: The convex hull of a random set of points. Biometrika 52, 331-343 (1965)
12. Finch, S., Hueter, I.: Random convex hulls: a variance revisited. Adv. Appl. Probab. 36, 981-986 (2004)
13. Groeneboom, P.: Limit theorems for convex hulls. Probab. Theory Relat. Fields 79, 327-368 (1988)
14. Gruber, P.M.: Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo, II. Ser., Suppl. 50, 189-216 (1997)
15. Henze, N.: Random triangles in convex regions. J. Appl. Prob. 20, 111-125 (1983)
16. Hsing, T.: On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab. 4, 478-493 (1994)
17. Hueter, I.: Limit theorems for the convex hull of random points in higher dimensions. Trans. Am. Math. Soc. 351, 4337-4363 (1999)
18. Hüsler, J.: On the convex hull of dependent random vectors. Rend. Circ. Mat. Palermo, II. Ser. Suppl. 41, 109-117 (1997)
19. McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179-184 (1970)
20. Reitzner, M.: Random points on the boundary of smooth convex bodies. Trans. Am. Math. Soc. 354, 2243-2278 (2002)
21. Reitzner, M.: Random polytopes and the Efron-Stein jackknife inequality. Ann. Probab. 31, 2136-2166 (2003)
22. Reitzner, M.: The combinatorial structure of random polytopes. Adv. Math. 191, 178-208 (2005)
23. Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitsth. Verw. Geb. 3, 138-147 (1964)
24. Rinott, Y.: On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55, 135-143 (1994)
25. Schneider, R.: Random approximation of convex sets. J. Microscopy 151, 211-227 (1988)
26. Schneider, R.: Discrete aspects of stochastic geometry. In: Goodman, J.E., O'Rourke, J. (eds.), Handbook of Discrete and Computational Geometry, Boca Raton: CRC Press 2004 (CRC Press Series on Discrete Mathematics and its Applications), pp. 255-278
27. Schneider, R., Weil, W.: Stochastische Geometrie. Stuttgart: Teubner Skripten zur Mathematischen Stochastik, 2000
28. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic geometry and its applications. Chichester: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, 1987
29. Schütt, C.: Random polytopes and affine surface area. Math. Nachr. 170, 227-249 (1994)
30. Vervaat, W.: Upper bounds for the distance in total variation between the binomial or negative binomial and the Poisson distribution. Statistica Neerlandica 23, 79-86 (1969)
31. Wieacker, J.A.: Einige Probleme der polyedrischen Approximation. Diplomarbeit, Freiburg im Breisgau 1978