The volume of simplices in high-dimensional Poisson–Delaunay tessellations

Gusakova, Anna; Thäle, Christoph

Annales Henri Lebesgue, Volume 4 (2021), pp. 121-153.

Typical weighted random simplices $Z_{\mu }$, $\mu \in (-2,\infty )$, in a Poisson–Delaunay tessellation in ${ℝ}^n$ are considered, where the weight is given by the $(\mu +1)$st power of the volume. As special cases this includes the typical ($\mu =-1$) and the usual volume-weighted ($\mu =0$) Poisson–Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_{\mu }$ satisfies a central limit theorem in high dimensions, that is, as $n\to \infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight $\mu =\mu \left(n\right)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $\mu$ also mod-$\varphi$ convergence and the large deviations behaviour of the logarithmic volume of $Z_{\mu }$ are investigated.

Nous considérons des simplexes aléatoires pondérés typiques $Z_{\mu }$, $\mu \in (-2,\infty )$, dans une tessellation de Poisson–Delaunay dans ${ℝ}^n$, où le poids est donné par la puissance $(\mu +1)$ du volume. Ceci inclut en particulier les simplexes de Poisson–Delaunay dans les cas typiques ($\mu =-1$) et pondérés par le volume ($\mu =0$). En prouvant des bornes précises sur les cumulants, nous montrons que le volume logarithmique de $Z_{\mu }$ satisfait un théorème central limite en grande dimension, i.e., lorsque $n\to \infty$. De plus, nous établissons des vitesses de convergence. Parallèlement, nous étudions les inégalités de concentration et les déviations modérées. Ce cadre permet au poids $\mu =\mu \left(n\right)$ de dépendre également de la dimension. Nous discutons séparément un certain nombre de cas particuliers. Lorsque $\mu$ est fixé, nous étudions également la convergence mod-$\varphi$ et les grandes déviations du volume logarithmique de $Z_{\mu }$.

Classification: 52A22, 60D05, 60F05, 60F10
Mots-clés : Berry–Esseen bound, central limit theorem, cumulants, high dimensions, mod-$\phi $ convergence, moderate deviations, large deviations, random simplex, Poisson–Delaunay tessellation, stochastic geometry

[AGBG + 19] Alonso-Gutiérrez, David; Besau, Florian; Grote, Julian; Kabluchko, Zakhar; Reitzner, Matthias; Thäle, Christoph; Vritsiou, Beatrice-Helen; Werner, Elisabeth M. Asymptotic normality for random simplices and convex bodies in high dimensions (2019) (https://arxiv.org/abs/1906.02471, to appear in Proceedings of the American Mathematical Society)

[AGPT18] Alonso-Gutiérrez, David; Prochno, Joscha; Thäle, Christoph Large deviations for high-dimensional random projections of ${\ell }_p^n$-balls, Adv. Appl. Math., Volume 99 (2018), pp. 1-35 | DOI | MR | Zbl

[AGPT19] Alonso-Gutiérrez, David; Prochno, Joscha; Thäle, Christoph Gaussian fluctuations for high-dimensional random projections of ${\ell }_p^n$-balls, Bernoulli, Volume 25 (2019) no. 4A, pp. 3139-3174 | DOI | MR | Zbl

[AS64] Abramowitz, Milton; Stegun, Irene A. Handbook of mathematical functions with formula, graphs and mathematical tables, Dover Publications, 1964 | Zbl

[BYY19] Błaszczyszyn, Bartłomiej; Yogeshwaran, Dhandapani; Yukich, Joseph E. Limit theory for geometric statistics of point processes having fast decay of correlations, Ann. Probab., Volume 47 (2019) no. 2, pp. 835-895 | DOI | MR | Zbl

[DE13a] Döring, Hanna; Eichelsbacher, Peter Moderate deviations for the determinant of Wigner matrices, Limit Theorems in Probability, Statistics and Number Theory (Springer Proceedings in Mathematics & Statistics), Volume 42, Springer, 2013, pp. 253-275 | DOI | MR | Zbl

[DE13b] Döring, Hanna; Eichelsbacher, Peter Moderate deviations via cumulants, J. Theor. Probab., Volume 26 (2013) no. 2, pp. 360-385 | DOI | MR | Zbl

[DKN15] Delbaen, Freddy; Kowalski, Emmanuel; Nikeghbali, Ashkan Mod-$\varphi$ convergence, Int. Math. Res. Not., Volume 11 (2015), pp. 3445-3485 | Zbl

[DT19] Dette, Holger; Tomecki, Dominik Determinants of block Hankel matrices for random matrix-valued measures, Stochastic Processes Appl., Volume 129 (2019) no. 12, pp. 5200-5235 | DOI | MR | Zbl

[DZ10] Dembo, Amir; Zeitouni, Ofer Large deviations techniques and applications, Stochastic Modelling and Applied Probability, 38, Springer, 2010 corrected reprint of the second (1998) edition | MR | Zbl

[EK20] Eichelsbacher, Peter; Knichel, Lukas Fine asymptotics for models with Gamma type moments (2020) (https://arxiv.org/abs/1710.06484, to appear in Random Matrices: Theory and Applications)

[ENR17] Edelsbrunner, Herbert; Nikitenko, Anton; Reitzner, Matthias Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions, Adv. Appl. Probab., Volume 49 (2017) no. 3, pp. 745-767 | DOI | MR | Zbl

[ERS15] Eichelsbacher, Peter; Raič, Martin; Schreiber, Tomasz Moderate deviations for stabilizing functionals in geometric probability, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 1, pp. 89-128 | DOI | Numdam | MR | Zbl

[FMN16] Féray, Valentin; Méliot, Pierre-Loïc; Nikeghbali, Ashkan Mod-$\varphi$ convergence. Normality zones and precise deviations. Normality zones and precise deviations, SpringerBriefs in Probability and Mathematical Statistics, Springer, 2016 | Zbl

[Fér18] Féray, Valentin Weighted dependency graphs, Electron. J. Probab., Volume 23 (2018), 93 | MR | Zbl

[GGZ19] Götze, Friedrich; Gusakova, Anna; Zaporozhets, Dmitry Random affine simplexes, J. Appl. Probab., Volume 56 (2019) no. 1, pp. 39-51 | DOI | MR | Zbl

[GKR17] Gantert, Nina; Kim, Steven Soojin; Ramanan, Kavita Large deviations for random projections of ${\ell }^p$ balls, Ann. Probab., Volume 45 (2017) no. 6B, pp. 4419-4476 | DOI | MR | Zbl

[GKT19] Grote, Julian; Kabluchko, Zakhar; Thäle, Christoph Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 16 (2019) no. 1, pp. 141-177 | DOI | MR | Zbl

[GT18a] Grote, Julian; Thäle, Christoph Concentration and moderate deviations for Poisson polytopes and polyhedra, Bernoulli, Volume 24 (2018) no. 4A, pp. 2811-2841 | DOI | MR | Zbl

[GT18b] Grote, Julian; Thäle, Christoph Gaussian polytopes: a cumulant-based approach, J. Complexity, Volume 47 (2018), pp. 1-41 | DOI | MR | Zbl

[Hei05] Heinrich, Lothar Large deviations of the empirical volume fraction for stationary Poisson grain models, Ann. Appl. Probab., Volume 15 (2005) no. 1A, pp. 392-420 | DOI | MR | Zbl

[Hof17] Hofer, Lisa A central limit theorem for vincular permutation patterns, Discrete Math. Theor. Comput. Sci., Volume 19 (2017) no. 2, 9 | MR | Zbl

[HS09] Heinrich, Lothar; Spiess, Malte Berry–Esseen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes, Lith. Math. J., Volume 49 (2009) no. 4, pp. 381-398 | DOI | Zbl

[JKN11] Jacod, Jean; Kowalski, Emmanuel; Nikeghbali, Ashkan Mod-Gaussian convergence: new limit theorems in probability and number theory, Forum Math., Volume 23 (2011) no. 4, pp. 3549-3587 | MR | Zbl

[Kla07] Klartag, Bo’az A central limit theorem for convex sets, Invent. Math., Volume 168 (2007) no. 1, pp. 91-131 | DOI | MR

[KPT19a] Kabluchko, Zakhar; Prochno, Joscha; Thäle, Christoph High-dimensional limit theorems for random vectors in ${\ell }_p^n$-balls, Commun. Contemp. Math., Volume 21 (2019) no. 1, 1750092 | MR | Zbl

[KPT19b] Kabluchko, Zakhar; Prochno, Joscha; Thäle, Christoph High-dimensional limit theorems for random vectors in ${\ell }_p^n$-balls. II (2019) (https://arxiv.org/abs/1906.03599 to appear in Communications in Contemporary Mathematics) | Zbl

[KTT19] Kabluchko, Zakhar; Temesvari, Daniel; Thäle, Christoph Expected intrinsic volumes and facet numbers of random beta-polytopes, Math. Nachr., Volume 292 (2019) no. 1, pp. 79-105 | DOI | MR | Zbl

[Mat82] Mathai, Arakaparampli M. On a conjecture in geometric probability regarding asymptotic normality of a random simplex, Ann. Probab., Volume 10 (1982) no. 1, pp. 247-251 | DOI | MR | Zbl

[Mil71] Miles, Roger E. Isotropic random simplices, Adv. Appl. Probab., Volume 3 (1971), pp. 353-382 | DOI | MR | Zbl

[Mor10] Mortici, Cristinel Very accurate estimates of the polygamma functions, Asymptotic Anal., Volume 68 (2010) no. 3, pp. 125-134 | DOI | MR | Zbl

[OLBC10] Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. NIST Handbook of Mathematical Functions, Cambridge University Press, 2010 | Zbl

[PPZ14] Paouris, Grigoris; Pivovarov, Peter; Zinn, Joel A central limit theorem for projections of the cube, Probab. Theory Relat. Fields, Volume 159 (2014) no. 3-4, pp. 701-719 | DOI | MR | Zbl

[PWZ17] Pan, Guangming; Wang, Shaochen; Zhou, Wang Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory, J. Math. Phys., Volume 58 (2017) no. 10, 103301 | MR | Zbl

[QV05] Qiu, Song-Liang; Vuorinen, Matti Some properties of the gamma and psi functions, with applications, Math. Comput., Volume 74 (2005) no. 250, pp. 723-742 | MR

[Rub77] Ruben, Harold The volume of a random simplex in an $n$-ball is asymptotically normal, J. Appl. Probab., Volume 14 (1977) no. 3, pp. 647-653 | DOI | MR | Zbl

[SS91] Saulis, Leonas; Statulevičius, Vytautas A. Limit Theorems for Large Deviations, Mathematics and its Applications (Soviet Series), 73, Kluwer Academic Publishers, 1991 (translated and revised from the 1989 Russian original) | MR

[ST16] Schulte, Matthias; Thäle, Christoph Cumulants on Wiener chaos: moderate deviations and the fourth moment theorem, J. Funct. Anal., Volume 270 (2016) no. 6, pp. 2223-2248 | DOI | MR | Zbl

[SW08] Schneider, Rolf; Weil, Wolfgang Stochastic and Integral Geometry, Probability and its Applications, Springer, 2008 | DOI | Zbl

[WW15] Whittaker, Edmund T.; Watson, George N. A Course of Modern Analysis, Cambridge University Press, 1915 | Zbl