## Annals of Probability

### A multivariate Gnedenko law of large numbers

Daniel Fresen

#### Abstract

We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach–Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case.

We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.

#### Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3051-3080.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991831

Digital Object Identifier
doi:10.1214/12-AOP804

Mathematical Reviews number (MathSciNet)
MR3127874

Zentralblatt MATH identifier
1293.60012

#### Citation

Fresen, Daniel. A multivariate Gnedenko law of large numbers. Ann. Probab. 41 (2013), no. 5, 3051--3080. doi:10.1214/12-AOP804. https://projecteuclid.org/euclid.aop/1378991831

#### References

• [1] Ball, K. (1992). Ellipsoids of maximal volume in convex bodies. Geom. Dedicata 41 241–250.
• [2] Ball, K. (1997). An elementary introduction to modern convex geometry. In Flavors of Geometry. Math. Sci. Res. Inst. Publ. 31 1–58. Cambridge Univ. Press, Cambridge.
• [3] Bárány, I. (2008). Random points and lattice points in convex bodies. Bull. Amer. Math. Soc. (N.S.) 45 339–365.
• [4] Bárány, I. and Larman, D. G. (1988). Convex bodies, economic cap coverings, random polytopes. Mathematika 35 274–291.
• [5] Bárány, I. and Vu, V. (2007). Central limit theorems for Gaussian polytopes. Ann. Probab. 35 1593–1621.
• [6] Baryshnikov, Y. and Vitale, R. (1994). Regular simplices and Gaussian samples. Discrete Comput. Geom. 11 141–147.
• [7] Bobkov, S. and Madiman, M. (2011). Concentration of the information in data with log-concave distributions. Ann. Probab. 39 1528–1543.
• [8] Bobkov, S. G. (2003). On concentration of distributions of random weighted sums. Ann. Probab. 31 195–215.
• [9] Dafnis, N., Giannopoulos, A. and Tsolomitis, A. (2009). Asymptotic shape of a random polytope in a convex body. J. Funct. Anal. 257 2820–2839.
• [10] Fresen, D. (2012). The floating body and the hyperplane conjecture. Arch. Math. (Basel) 98 389–397.
• [11] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44 423–453.
• [12] Goodman, V. (1988). Characteristics of normal samples. Ann. Probab. 16 1281–1290.
• [13] Gruber, P. M. (1993). Aspects of approximation of convex bodies. In Handbook of Convex Geometry, Vols. A, B 319–345. North-Holland, Amsterdam.
• [14] Klartag, B. and Milman, V. D. (2005). Geometry of log-concave functions and measures. Geom. Dedicata 112 169–182.
• [15] Koldobsky, A. (2005). Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs 116. Amer. Math. Soc., Providence, RI.
• [16] Lovász, L. and Vempala, S. (2007). The geometry of logconcave functions and sampling algorithms. Random Structures Algorithms 30 307–358.
• [17] Matoušek, J. (2002). Lectures on Discrete Geometry. Graduate Texts in Mathematics 212. Springer, New York.
• [18] Milman, V. D. and Schechtman, G. (1986). Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin.
• [19] Naor, A. and Romik, D. (2003). Projecting the surface measure of the sphere of $l_{p}^{n}$. Ann. Inst. Henri Poincaré Probab. Stat. 39 241–261.
• [20] Pisier, G. (1989). The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics 94. Cambridge Univ. Press, Cambridge.
• [21] Raynaud, H. (1970). Sur l’enveloppe convexe des nuages de points aléatoires dans $R^{n}$. I. J. Appl. Probab. 7 35–48.
• [22] Schechtman, G. and Zinn, J. (1990). On the volume of the intersection of two $L^{n}_{p}$ balls. Proc. Amer. Math. Soc. 110 217–224.
• [23] Schneider, R. (1987). Polyhedral approximation of smooth convex bodies. J. Math. Anal. Appl. 128 470–474.
• [24] Schütt, C. and Werner, E. (1990). The convex floating body. Math. Scand. 66 275–290.