Extremal random beta polytopes

E. N. Simarova

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 30, Tome 501 (2021), pp. 276-301

Résumé

The convex hull of several i.i.d. beta distributed random vectors in $\mathbb R^d$ is called the random beta polytope. Recently, the expected values of their intrinsic volumes, number of faces, normal and tangent angles and other quantities have been calculated, explicitly and asymptotically. In this paper, we aim to investigate the asymptotic behavior of the beta polytopes with extremal intrinsic volumes. We suggest a conjecture and solve it in dimension $2$. To this end, we obtain some general limit relation for a wide class of $U$-$\max$ statistics whose kernels include the perimeter and the area of the convex hull of the arguments.

Bibliographie

[1] A. D. Barbour, L. Holst, S. Janson, Poisson Approximation, Oxford University Press, London, 1992 | Zbl

[2] J. Berman, K. Hanes, “Volumes of polyhedra inscribed in the unit sphere in $E^3$”, Mathematische Annalen, 188:1 (1970), 78–84 | DOI | MR | Zbl

[3] P. R. Halmos, “The theory of unbiased estimation”, Ann. Math. Statist., 17 (1946), 34–43 | DOI | Zbl

[4] N. Henze, T. Klein, “The limit distribution of the largest interpoint distance from a symmetric Kotz sample”, J. Multivariate Anal., 57 (1996), 228–239 | DOI | MR | Zbl

[5] W. Hoeffding, “A class of statistics with asymptotically normal distribution”, Ann. Math. Statist., 19 (1948), 293–325 | DOI | Zbl

[6] Á. Horváth, Z. Lángi, “Maximum volume polytopes inscribed in the unit sphere”, Monatshefte für Mathematik, 281:2 (2016), 341–354

[7] S. R. Jammalamadaka, S. Janson, “Asymptotic distribution of the maximum interpoint distance in a sample of random vectors with a spherically symmetric distribution”, Ann. Appl. Probab., 25:6 (2015), 3571–3591 | DOI | MR | Zbl

[8] Z. Kabluchko, “Angles of random simplices and face numbers of random polytopes”, Adv. Math., 380 (2021), 107612 | DOI | Zbl

[9] Z. Kabluchko, D. Temesvari, C. Thäle, “Expected intrinsic volumes and facet numbers of random beta-polytopes”, Math. Nachr., 292 (2019), 79–105 | DOI | MR | Zbl

[10] Z. Kabluchko, C. Thäle, D. Zaporozhets, “Beta polytopes and Poisson polyhedra: f-vectors and angles”, Adv. Math., 374 (2020), 107333 | DOI | MR | Zbl

[11] E. V. Koroleva, Ya. Yu. Nikitin, “$U$-max-statistics and limit theorems for perimeters and areas of random polygons”, J. Multivariate Anal., 127 (2014), 99–111 | DOI

[12] W. Lao, Some weak limit laws for the diameter of random point sets in bounded regions, Ph.D. Thesis, Karlsruhe, 2010

[13] W. Lao, M. Mayer, “$U$-max-statistics”, J. Multivariate Anal., 99 (2008), 2039–2052 | DOI | MR | Zbl

[14] A. J. Lee, $U$-statistics: Theory and Practice, Routledge, 2019

[15] P. C. Matthews, A. L. Rukhin, “Asymptotic distribution of the normal sample range”, Ann. Appl. Probab., 3 (1993), 454–466 | DOI | MR

[16] M. Mayer, Random Diameters and Other $U$-max-Statistics, Ph.D. Thesis, Bern University, 2008 | Zbl

[17] M. Mayer, I. Molchanov, “Limit theorems for the diameter of a random sample in the unit ball”, Extremes, 10 (2007), 151–174 | DOI | MR

[18] Ya. Yu. Nikitin, T. A. Polevaya, “Predelnye teoremy dlya ploschadei i perimetrov sluchainykh vpisannykh i opisannykh mnogougolnikov”, Zap. nauchn. semin. POMI, 486, 2019, 200–213

[19] Ya. Yu. Nikitin, E. N. Simarova, Generalized Limit Theorems For U-max Statistics, preprint

[20] F. B. Silverman, T. Brown, “Short distances, flat triangles, and Poisson limits”, J. Appl. Probab., 15 (1978), 815–825 | DOI | MR | Zbl

[21] E. N. Simarova, “Predelnye teoremy dlya obobschennykh perimetrov sluchainykh vpisannykh mnogougolnikov. I”, Vestnik Sankt-Peterburgskogo universiteta. Matematika. Mekhanika. Astronomiya, 7 (65):4 (2020), 678–687 | MR | Zbl

[22] E. N. Simarova, “Predelnye teoremy dlya obobschennykh perimetrov sluchainykh vpisannykh mnogougolnikov. II”, Vestnik Sankt-Peterburgskogo universiteta. Matematika. Mekhanika. Astronomiya, 8 (66):1 (2021), 78–85 | Zbl