## 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

### Keywords:

Random polytopes; Gaussian distribution; dependency graph
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### References:

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