Concentration properties of functionals of general Poisson processes are studied. Using a modified $Phi$-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentr
ation inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well.
The typical cell of a Voronoi tessellation generated by $n+1$ uniformly distributed random points on the $d$-dimensional unit sphere $mathbb S^d$ is studied. Its $f$-vector is identified in distribution with the $f$-vector of a beta polytope generate
d by $n$ random points in $mathbb R^d$. Explicit formulae for the expected $f$-vector are provided for any $d$ and the low-dimensional cases $din{2,3,4}$ are studied separately. This implies an explicit formula for the total number of $k$-dimensional faces in the spherical Voronoi tessellation as well.
In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations pri
nciple when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guedon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thale: High-dimensional limit theorems for random vectors in $ell_p^n$-balls, Commun. Contemp. Math. (2019)].
The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of
the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to $p$-affine surface areas.
The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $dgeq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studi
ed using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $kin{1,ldots,d-1}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.
The unit ball $B_p^n(mathbb{R})$ of the finite-dimensional Schatten trace class $mathcal S_p^n$ consists of all real $ntimes n$ matrices $A$ whose singular values $s_1(A),ldots,s_n(A)$ satisfy $s_1^p(A)+ldots+s_n^p(A)leq 1$, where $p>0$. Saint Raymon
d [Studia Math. 80, 63--75, 1984] showed that the limit $$ lim_{ntoinfty} n^{1/2 + 1/p} big(text{Vol}, B_p^n(mathbb{R})big)^{1/n^2} $$ exists in $(0,infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields. A similar result is obtained for complex Schatten balls. As an application we compute the precise asymptotic volume ratio of the Schatten $p$-balls, as $ntoinfty$, thereby extending Saint Raymonds estimate in the case of the nuclear norm ($p=1$) to the full regime $1leq p leq infty$ with exact limiting behavior.
We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschlager for c
lassical $ell_p$-balls [Schechtman and Schmuckenschlager, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.
Let $X_1,ldots,X_n$ be i.i.d. random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,beta} (x) = text{const} cdot (1-|x|^2)^{beta}, quad |x|leq 1, quad text{(the beta case)} $$ an
d $$ tilde f_{d,beta} (x) = text{const} cdot (1+|x|^2)^{-beta}, quad xinmathbb{R}^d, quad text{(the beta case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when $betadownarrow -1$, respectively $beta uparrow +infty$, we can also cover the models in which $X_1,ldots,X_n$ are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of $pm X_1,ldots,pm X_n$ and $0,X_1,ldots,X_n$. One of the main tools used in the proofs is the Blaschke-Petkantschin formula.
The random convex hull of a Poisson point process in $mathbb{R}^d$ whose intensity measure is a multiple of the standard Gaussian measure on $mathbb{R}^d$ is investigated. The purpose of this paper is to invent a new viewpoint on these Gaussian polyt
opes that is based on cumulants and the general large deviation theory of Saulis and Statuleviv{c}ius. This leads to new and powerful concentration inequalities, moment bounds, Marcinkiewicz-Zygmund-type strong laws of large numbers, central limit theorems and moderate deviation principles for the volume and the face numbers. Corresponding results are also derived for the empirical measures induced by these key geometric functionals, taking thereby care of their spatial profiles.
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 probabiliti
es 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.
