Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGaussian polytopes: a cumulant-based approach
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 polytopes 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.
rate research
Read More
A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schlafli random cone of a random conical tessellation generated by $n$ independent and uniformly distributed random linear hyperplanes in $mathbb{R}^{d+1}$ weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $mathbb{R}^d$, as $n to infty$.
Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Furedi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.
Fix a space dimension $dge 2$, parameters $alpha > -1$ and $beta ge 1$, and let $gamma_{d,alpha, beta}$ be the probability measure of an isotropic random vector in $mathbb{R}^d$ with density proportional to begin{align*} ||x||^alpha, expleft(-frac{|x|^beta}{beta}right), qquad xin mathbb{R}^d. end{align*} By $K_lambda$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in $mathbb{R}^d$ with intensity measure $lambdagamma_{d,alpha,beta}$, $lambda>0$. We establish that the scaling limit of the boundary of $K_lambda$, as $lambda rightarrow infty$, is given by a universal `festoon of piecewise parabolic surfaces, independent of $alpha$ and $beta$. Moreover, we state a list of other large scale asymptotic results, including expectation and variance asymptotics, central limit theorems, concentration inequalities, Marcinkiewicz-Zygmund-type strong laws of large numbers, as well as moderate deviation principles for the intrinsic volumes and face numbers of $K_lambda$.
We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size. We establish an explicit asymptotic formula that is valid whenever d/n tends to zero. We also obtain the asymptotic value when d is close to n.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
