Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T09:09:31.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Gaussian polytopes: variances and limit theorems

Part of: Combinatorial probability Geometric probability and stochastic geometry General convexity Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

Daniel Hug  and
Matthias Reitzner
Daniel Hug*
Affiliation:
Albert-Ludwigs-Universität Freiburg
Matthias Reitzner*
Affiliation:
Technische Universität Wien
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstr. 1, D-79104 Freiburg i. Br., Germany. Email address: [email protected]
∗∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The convex hull of n independent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith intrinsic volume of a Gaussian polytope in ℝd, d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the ith intrinsic volume of a Gaussian polytope as n→∞.

Keywords

Random pointconvex hullf-vectorintrinsic volumegeometric probabilitynormal distributionGaussian samplestochastic geometryvariancelaw of large numberslimit theorem

MSC classification

Primary: 52A22: Random convex sets and integral geometry 60D05: Geometric probability and stochastic geometry
Secondary: 60C05: Combinatorial probability 62H10: Distribution of statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 297 - 320
Copyright
Copyright © Applied Probability Trust 2005 

References

Affentranger, F. (1991). The convex hull of random points with spherically symmetric distributions. Rend. Sem. Mat. Univ. Politec. Torino 49, 359–383.Google Scholar
Affentranger, F. and Schneider, R. (1992). Random projections of regular simplices. Discrete Comput. Geom. 7, 219–226.Google Scholar
Baryshnikov, Y. M. and Vitale, R. A. (1994). Regular simplices and Gaussian samples. Discrete Comput. Geom. 11, 141–147.Google Scholar
Efron, B. and Stein, C. (1981). The Jackknife estimate of variance. Ann. Statist. 9, 586–596.Google Scholar
Geffroy, J. (1961). Localisation asymptotique du polyèdre d' appui d' un échantillon Laplacien à k dimensions. Publ. Inst. Statist. Univ. Paris 10, 213–228.Google Scholar
Henze, N. and Klein, T. (1996). The limit distribution of the largest interpoint distance from a symmetric Kotz sample. J. Multivariate Anal. 57, 228–239.Google Scholar
Hueter, I. (1994). The convex hull of a normal sample. Adv. Appl. Prob. 26, 855–875.Google Scholar
Hueter, I. (1999). Limit theorems for the convex hull of random points in higher dimensions. Trans. Amer. Math. Soc. 351, 4337–4363.CrossRefGoogle Scholar
Hug, D., Munsonius, G. O. and Reitzner, M. (2004). Asymptotic mean values of Gaussian polytopes. Beiträge Algebra Geom. 45, 531–548.Google Scholar
Hüsler, J. (1997). Range of bivariate normal random vectors. Rend. Circ. Mat. Palermo (2) Suppl. 50, 229–234.Google Scholar
Jensen, E. B. V. (1998). Local Stereology (Adv. Ser. Statist. Sci. Appl. Prob. 5). World Scientific, River Edge, NJ.Google Scholar
Jensen, E. B. V. and Kiêu, K. (1992). A new integral geometric formula of the Blaschke–Petkantschin type. Math. Nachr. 156, 57–74.CrossRefGoogle Scholar
Kendall, M. G. and Stuart, A. (1965). The Advanced Theory of Statistics: Distribution Theory, Vol. 1, 3rd edn. Griffin, London.Google Scholar
Mardia, K. V. (1965). Tippet's formulas and other results on sample range and extremes. Ann. Inst. Statist. Math. 17, 85–91.Google Scholar
Massé, B. (2000). On the LLN for the number of vertices of a random convex hull. Adv. Appl. Prob. 32, 675–681.Google Scholar
Matthews, P. C. and Rukhin, A. L. (1993). Aysmptotic distribution of the normal sample range. Ann. Appl. Prob. 3, 454–466.Google Scholar
Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
Patel, J. K. and Read, C. B. (1996). Handbook of the Normal Distribution (Statist. Textbook Monogr. 40). Marcel Dekker, New York.Google Scholar
Raynaud, H. (1970). Sur l' enveloppe convexe des nuages de points aléatoires dans R n . I. J. Appl. Prob. 7, 35–48.Google Scholar
Reitzner, M. (2003). Random polytopes and the Efron–Stein Jackknife inequality. Ann. Prob. 31, 2136–2166.Google Scholar
Reitzner, M. (2005). The combinatorial structure of random polytopes. Adv. Math. 191, 178–208.Google Scholar
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.Google Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schneider, R. (1997). Discrete aspects of stochastic geometry. In Handbook of Discrete and Computational Geometry, eds Goodman, J. E. and O' Rourke, J., CRC, Boca Raton, FL, pp. 255–278.Google Scholar
Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.Google Scholar
Vershik, A. M. and Sporyshev, P. V. (1992). Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem. Select. Math. Soviet. 11, 181–201.Google Scholar
Wendel, J. G. (1962). A problem in geometric probability. Math. Scand. 11, 109–111.Google Scholar