## 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:

 [1] Affentranger, F. (1991). The convex hull of random points with spherically symmetric distributions. Rend. Sem. Mat. Univ. Politec. Torino 49 359–383. · Zbl 0774.60015 [2] Affentranger, F. and Schneider, R. (1992). Random projections of regular simplices. Discrete Comp. Geom. 7 219–226. · Zbl 0751.52002 [3] Baldi, P. and Rinott, Y. (1989). On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17 1646–1650. · Zbl 0691.60020 [4] Bárány, I. and Dalla, L. (1997). Few points to generate a random polytope. Mathematika 44 325–331. · Zbl 0902.52002 [5] Bárány, I. and Reitzner, M. (2006). Central limit theorems for random polytopes in convex polytopes. Probab. Theory Related Fields 133 483–507. · Zbl 1081.60008 [6] Baryshnikov, Y. M. and Vitale, R. A. (1994). Regular simplices and Gaussian samples. Discrete Comp. Geom. 11 141–147. · Zbl 0795.52002 [7] Cabo, A. J. and Groeneboom, P. (1994). Limit theorems for functionals of convex hulls. Probab. Theory Related Fields 100 31–55. · Zbl 0808.60019 [8] Groeneboom, P. (1988). Limit theorems for convex hulls. Probab. Theory Related Fields 79 327–368. · Zbl 0635.60012 [9] Hsing, T. (1994). On the asymptotic distribution of th area outside a random convex hull in a disk. Ann. Appl. Probab. 4 478–493. · Zbl 0806.60004 [10] Hueter, I. (1994). The convex hull of a normal sample. Adv. in Appl. Probab. 26 855–875. JSTOR: · Zbl 0815.60013 [11] Hueter, I. (1999). Limit theorems for the convex hull of random points in higher dimensions. Trans. Amer. Math. Soc. 351 4337–4363. JSTOR: · Zbl 0944.60018 [12] Hug, D., Munsonius, G. O. and Reitzner, M. (2004). Asymptotic mean values of Gaussian polytopes. Beiträge Algebra Geom. 49 531–548. · Zbl 1082.52003 [13] Hug, D. and Reitzner, M. (2005). Gaussian polytopes: Variances and limit theorems. Adv. in Appl. Probab. 37 297–320. · Zbl 1089.52003 [14] Raynaud, H. (1970). Sur l’enveloppe convexe des nuages de points aléatoires dans $$\R^n$$. J. Appl. Probab. 7 35–48. JSTOR: · Zbl 0192.53602 [15] Reitzner, M. (2005). Central limit theorems for random polytopes. Probab . Theory Related Fields 133 483–507. · Zbl 1081.60008 [16] Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von $$n$$ zufällig gewählten Punkten. Z. Wahrsch. Verw. Gebiete 2 75–84. · Zbl 0118.13701 [17] Rinott, Y. (1994). On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55 135–143. · Zbl 0821.60037 [18] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 583–602. Univ. California Press, Berkeley. · Zbl 0278.60026 [19] Vu, V. H. (2005). Sharp concentration of random polytopes. Geom. Funct. Anal. 15 1284–1318. · Zbl 1094.52002 [20] Vu, V. H. (2006). Central limit theorems for random polytopes in a smooth convex set. Adv. in Math. 207 221–243. · Zbl 1111.52010 [21] Weil, W. and Wieacker, J. (1993). Stochastic geometry. In Handbook of Convex Geometry A , B 1391–1438. North-Holland, Amsterdam. · Zbl 0788.52002
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