Concentration and moderate deviations for Poisson polytopes and polyhedra

Abstract

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the d-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.

Gaussian polytopes: A cumulant-based approach

“…The theorems we have seen in this section (and also those in Section 3 below) are the clear analogues for Gaussian polytopes of the results recently derived in our paper [26], where we considered random polytopes that arise as convex hulls of a homogeneous Poisson point process in the d-dimensional unit ball. Moreover, also the principal technique we use, based on sharp bounds for cumulants in conjunction with the large deviation theory from [41], parallels that in [26]. However, we emphasize at this point that besides of these conceptual similarities, the further details and arguments differ considerably and require much more technical effort as well as a number of new ideas compared to [26].…”

“…Moreover, also the principal technique we use, based on sharp bounds for cumulants in conjunction with the large deviation theory from [41], parallels that in [26]. However, we emphasize at this point that besides of these conceptual similarities, the further details and arguments differ considerably and require much more technical effort as well as a number of new ideas compared to [26]. This is basically due to the fact that, in contrast to random polytopes in the unit ball, Gaussian polytopes in R d grow unboundedly in all directions.…”

Normal approximation for stabilizing functionals

“…The following result generalizes [9, (2.7)] to non-spherical compact sets, with arguments similar to Lemma A1 from [16]. The proof is in the appendix.…”

Limit theorems for random simplices in high dimensions

“…in the Gaussian case and Remark 3.6. Starting with the cumulant bounds presented in Theorem 3.1 one can also derive (i) concentration inequalities, (ii) bound for moments of all orders, (iii) Cramér-Petrov type results concerning the relative error in the central limit theorem, (iv) strong laws of large numbers for the random variables L n,r from the results presented in [33, Chapter 2] (see also [13,14]).…”

“…Proposition 4. 13. Consider the Beta model with parameter ν > 0 or the spherical model (in which case ν = 0) and let m n be the same as in Proposition 4.7.…”