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Central limit theorems for Gaussian polytopes. (English) Zbl 1124.60014

Ann. Probab. 35, No. 4, 1593-1621 (2007); correction ibid. 36, No. 5, 1998 (2008).
Let \(\Psi_d=\Psi\) denote the standard normal distribution on \(\mathbb R^d\), \(d\geq2\), its density function is \(\psi_d=\psi=\exp\{-x^2/2\}/(2\pi)^{d/2}\), where \(x^2=| x| ^2\) is the square of the Euclidean norm of \(x\in\mathbb R^d\). Choose a set \(X_n=\{x_1,..., x_n\}\) of random independent points from \(\mathbb R^d\) according to the normal distribution \(\Psi\). The convex hull of these points, \(K_n\), is the Gaussian polytope. This is one of the central models in the theory of random polytopes (see A. Rényi, and R. Sulanke [Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 75–84 (1963; Zbl 0118.13701)]). The main goal of this theory is to investigate the distributions of the key functionals of random polytopes. It is a natural and important conjecture in the theory of random polytopes that the key functionals of \(K_n\) satisfy the central limit theorem, as \(n\) tends to infinity. However, early results are very far from a possible answer of this question and mostly focused on expectations. Let \(\text{Vol}(K_n)\) and \(f_s(K_n)\) be the volume and number of faces of dimension \(s\), respectively. H. Raynaud [J. Appl. Probab. 7, 35–48 (1970; Zbl 0192.53602)] computed \(Ef_{d-1}(K_n)\) in all dimensions. Recently, D. Hug and M. Reitzner [Adv. Appl. Probab. 37, No. 2, 297–320 (2005; Zbl 1089.52003)] obtained an estimate for the variance. An upper bound for the variance of \(\text{Vol}(K_n)\) is given by Hug and Reitzner (loc. cit.). Therefore, the conjecture has been open for several decades and very few partial results have been proved.
In this paper, the authors develop a general framework which enables to confirm this conjecture for the volume of \(K_n\) and the number of faces \(f_s(K_n)\) of any dimension \(s\). The framework of the present paper makes use of ideas from I. Bárány and M. Reitzner [“Central limit theorems for random polytopes in convex polytopes,” manuscript (2005)], M. Reitzner [Probab. Theory Relat. Fields 133, No. 4, 483–507 (2005; Zbl 1081.60008)] and V. H. Vu [Geom. Funct. Anal. 15, No. 6, 1284–1318 (2005; Zbl 1094.52002) and Adv. Math. 207, No. 1, 221–243 (2006; Zbl 1111.52010)].

MSC:

60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
60C05 Combinatorial probability
60F15 Strong limit theorems
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References:

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