Asymptotic normality for random simplices and convex bodies in high dimensions

Alonso-Gutiérrez, D.; Besau, F.; Grote, J.; Kabluchko, Z.; Reitzner, M.; Thäle, C.; Vritsiou, B.-H.; Werner, E.

doi:10.1090/proc/15232

Asymptotic normality for random simplices and convex bodies in high dimensions
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by D. Alonso-Gutiérrez, F. Besau, J. Grote, Z. Kabluchko, M. Reitzner, C. Thäle, B.-H. Vritsiou and E. Werner

Proc. Amer. Math. Soc. 149 (2021), 355-367

DOI: https://doi.org/10.1090/proc/15232

Published electronically: October 9, 2020

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Abstract:

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb {R}^n$ are obtained in the high-dimensional regime, that is, as $n\to \infty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is also established for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $\ell _p$-ball. In particular, this includes the cone and the uniform probability measure.

References

Milton Abramowitz (ed.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1965. Superintendent of Documents. MR 0177136
David Alonso-Gutiérrez, Joscha Prochno, and Christoph Thäle, Gaussian fluctuations for high-dimensional random projections of $\ell _p^n$-balls, Bernoulli 25 (2019), no. 4A, 3139–3174. MR 4003577, DOI 10.3150/18-BEJ1084
Imre Bárány, Random polytopes, convex bodies, and approximation, Stochastic geometry, Lecture Notes in Math., vol. 1892, Springer, Berlin, 2007, pp. 77–118. MR 2327291, DOI 10.1007/978-3-540-38175-4_{2}
Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor, A probabilistic approach to the geometry of the $l^n_p$-ball, Ann. Probab. 33 (2005), no. 2, 480–513. MR 2123199, DOI 10.1214/009117904000000874
Richard F. Bass, Stochastic processes, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 33, Cambridge University Press, Cambridge, 2011. MR 2856623, DOI 10.1017/CBO9780511997044
Peter Eichelsbacher and Lukas Knichel, Fine asymptotics for models with Gamma type moments, arXiv:1710.06484.
Julian Grote, Zakhar Kabluchko, and Christoph Thäle, Limit theorems for random simplices in high dimensions, ALEA Lat. Am. J. Probab. Math. Stat. 16 (2019), no. 1, 141–177. MR 3903027, DOI 10.30757/alea.v16-06
Daniel Hug, Random polytopes, Stochastic geometry, spatial statistics and random fields, Lecture Notes in Math., vol. 2068, Springer, Heidelberg, 2013, pp. 205–238. MR 3059649, DOI 10.1007/978-3-642-33305-7_{7}
Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169, DOI 10.1007/978-1-4757-4015-8
A. M. Mathai, On a conjecture in geometric probability regarding asymptotic normality of a random simplex, Ann. Probab. 10 (1982), no. 1, 247–251. MR 637392, DOI 10.1214/aop/1176993929
Hiroshi Maehara, On random simplices in product distributions, J. Appl. Probab. 17 (1980), no. 2, 553–558. MR 568966, DOI 10.1017/s0021900200047380
R. E. Miles, Isotropic random simplices, Advances in Appl. Probability 3 (1971), 353–382. MR 309164, DOI 10.2307/1426176
Hoi H. Nguyen and Van Vu, Random matrices: law of the determinant, Ann. Probab. 42 (2014), no. 1, 146–167. MR 3161483, DOI 10.1214/12-AOP791
Hoi H. Nguyen and Van H. Vu, Normal vector of a random hyperplane, Int. Math. Res. Not. IMRN 6 (2018), 1754–1778. MR 3800634, DOI 10.1093/imrn/rnw273
Grigoris Paouris and Peter Pivovarov, A probabilistic take on isoperimetric-type inequalities, Adv. Math. 230 (2012), no. 3, 1402–1422. MR 2921184, DOI 10.1016/j.aim.2012.03.019
Grigoris Paouris and Peter Pivovarov, Small-ball probabilities for the volume of random convex sets, Discrete Comput. Geom. 49 (2013), no. 3, 601–646. MR 3038532, DOI 10.1007/s00454-013-9492-2
G. Paouris, P. Pivovarov, and P. Valettas, Gaussian convex bodies: a non-asymptotic approach, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 457 (2017), no. Veroyatnost′i Statistika. 25, 286–316; English transl., J. Math. Sci. (N.Y.) 238 (2019), no. 4, 537–559. MR 3723587, DOI 10.1007/s10958-019-04256-3
Matthias Reitzner, Random polytopes, New perspectives in stochastic geometry, Oxford Univ. Press, Oxford, 2010, pp. 45–76. MR 2654675
Harold Ruben, The volume of a random simplex in an $n$-ball is asymptotically normal, J. Appl. Probability 14 (1977), no. 3, 647–653. MR 448479, DOI 10.2307/3213472
G. Schechtman and J. Zinn, On the volume of the intersection of two $L^n_p$ balls, Proc. Amer. Math. Soc. 110 (1990), no. 1, 217–224. MR 1015684, DOI 10.1090/S0002-9939-1990-1015684-0