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Item type:Publication, Access status: with files , The radial spanning tree in hyperbolic space: degree and edge-length*(Institute of Mathematical Statistics, 2025-08-07); ; ; Consider a stationary Poisson process η in a d-dimensional hyperbolic space of constant curvature −κ and let the points of η together with a fixed origin o be the vertices of a graph. Connect each point x ∈ η with its radial nearest neighbour, which is the hyperbolically nearest vertex to x that is closer to o than x. This construction gives rise to the hyperbolic radial spanning tree, whose geometric properties are in the focus of this paper. In particular, the degree of the origin is studied. For increasing balls around o as observation windows, expectation and variance asymptotics as well as a quantitative central limit theorem for a class of edge-length functionals are derived. The results are contrasted with those for the Euclidean radial spanning tree.Publicationtype: Journal Article17 24 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Limit theorems for general functionals of Brownian local times(Institute of Mathematical Statistics, 2024-09-17); 8 31 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Lower bounds for variances of poisson functionalsLower bounds for variances are often needed to derive central limit theorems. In this paper, we establish a lower bound for the variance of Poisson functionals that uses the difference operator of Malliavin calculus. Poisson functionals, i.e. random variables that depend on a Poisson process, are frequently studied in stochastic geometry. We apply our lower variance bound to statistics of spatial random graphs, the Lp surface area of random polytopes and the volume of excursion sets of Poisson shot noise processes. Thereby we do not only bound variances from below but also show positive definiteness of asymptotic covariance matrices and provide associated results on the multivariate normal approximation.Publicationtype: Journal Article13 55 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Moderate deviations on poisson chaosThis paper deals with U-statistics of Poisson processes and multiple Wiener-Itô integrals on the Poisson space. Via sharp bounds on the cumulants for both classes of random variables, moderate deviation principles, concentration inequalities and normal approximation bounds with Cramér correction are derived. It is argued that the results obtained in this way are in a sense best possible and cannot be improved systematically. Applications in stochastic geometry and to functionals of OrnsteinUhlenbeck-Lévy processes are investigated.Publicationtype: Journal Article10 22 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes(University of Washington, Mathematics Department, 2022-06); ; This paper deals with the union set of a stationary Poisson process of cylinders in Rn having an (n − m)-dimensional base and an m-dimensional direction space, where m ∈ 0, 1, …, n − 1 and n ≥ 2. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or m-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.Publicationtype: Journal Article74 127 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Poisson approximation with applications to stochastic geometryThis article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.Publicationtype: Journal Article78