Random polytopes: Central limit theorems for intrinsic volumes

Thäle, Christoph; Turchi, Nicola; Wespi, Florian

doi:10.1090/proc/14000

Random polytopes: Central limit theorems for intrinsic volumes
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by Christoph Thäle, Nicola Turchi and Florian Wespi PDF

Proc. Amer. Math. Soc. 146 (2018), 3063-3071 Request permission

Abstract:

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein’s method from probability theory.

References

Imre Bárány, Random points and lattice points in convex bodies, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 3, 339–365. MR 2402946, DOI 10.1090/S0273-0979-08-01210-X
I. Bárány, F. Fodor, and V. Vígh, Intrinsic volumes of inscribed random polytopes in smooth convex bodies, Adv. in Appl. Probab. 42 (2010), no. 3, 605–619. MR 2779551, DOI 10.1239/aap/1282924055
Imre Bárány and Leoni Dalla, Few points to generate a random polytope, Mathematika 44 (1997), no. 2, 325–331. MR 1600549, DOI 10.1112/S0025579300012638
I. Bárány and D. G. Larman, Convex bodies, economic cap coverings, random polytopes, Mathematika 35 (1988), no. 2, 274–291. MR 986636, DOI 10.1112/S0025579300015266
Imre Bárány and Christoph Thäle, Intrinsic volumes and Gaussian polytopes: the missing piece of the jigsaw, Doc. Math. 22 (2017), 1323–1335. MR 3722561
Pierre Calka, Tomasz Schreiber, and J. E. Yukich, Brownian limits, local limits and variance asymptotics for convex hulls in the ball, Ann. Probab. 41 (2013), no. 1, 50–108. MR 3059193, DOI 10.1214/11-AOP707
Sourav Chatterjee, A new method of normal approximation, Ann. Probab. 36 (2008), no. 4, 1584–1610. MR 2435859, DOI 10.1214/07-AOP370
Raphaël Lachièze-Rey and Giovanni Peccati, New Berry-Esseen bounds for functionals of binomial point processes, Ann. Appl. Probab. 27 (2017), no. 4, 1992–2031. MR 3693518, DOI 10.1214/16-AAP1218
R. Lachièze-Rey, M. Schulte and J.E. Yukich : Normal approximation for stabilizing functionals. (2017), arXiv: 1702.00726.
John Pardon, Central limit theorems for uniform model random polygons, J. Theoret. Probab. 25 (2012), no. 3, 823–833. MR 2956214, DOI 10.1007/s10959-010-0335-2
Matthias Reitzner, Random polytopes and the Efron-Stein jackknife inequality, Ann. Probab. 31 (2003), no. 4, 2136–2166. MR 2016615, DOI 10.1214/aop/1068646381
Matthias Reitzner, Central limit theorems for random polytopes, Probab. Theory Related Fields 133 (2005), no. 4, 483–507. MR 2197111, DOI 10.1007/s00440-005-0441-8
Matthias Reitzner, Random polytopes, New perspectives in stochastic geometry, Oxford Univ. Press, Oxford, 2010, pp. 45–76. MR 2654675
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
Tomasz Schreiber, Variance asymptotics and central limit theorems for volumes of unions of random closed sets, Adv. in Appl. Probab. 34 (2002), no. 3, 520–539. MR 1929596, DOI 10.1239/aap/1033662164
Carsten Schütt and Elisabeth Werner, The convex floating body, Math. Scand. 66 (1990), no. 2, 275–290. MR 1075144, DOI 10.7146/math.scand.a-12311
V. H. Vu, Sharp concentration of random polytopes, Geom. Funct. Anal. 15 (2005), no. 6, 1284–1318. MR 2221249, DOI 10.1007/s00039-005-0541-8
Van Vu, Central limit theorems for random polytopes in a smooth convex set, Adv. Math. 207 (2006), no. 1, 221–243. MR 2264072, DOI 10.1016/j.aim.2005.11.011