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Item type:Publication, Access status: without files , First passage percolation on Erdős–Rényi graphs with general weightsWe consider first passage percolation on the Erdős–Rényi graph with n vertices in which each pair of distinct vertices is connected independently by an edge with probability λ/n for some λ>1. The edges of the graph are given nonnegative i.i.d. weights with a nondegenerate distribution such that the probability of zero is not too large. We consider the paths with small total weight between two distinct typical vertices and analyse the joint behaviour of the numbers of edges on such paths, the so-called hopcounts, and the total weights of these paths. For n→∞, we show that, after a suitable transformation, the pairs of hopcounts and total weights of these paths converge in distribution to a Cox process, that is, a Poisson process with a random intensity measure. The random intensity measure is controlled by two independent random variables, whose distribution is the solution of a distributional fixed point equation and is related to branching processes. For nonarithmetic and arithmetic edge-weight distributions we observe different behaviour. In particular, we derive the limiting distribution for the minimal total weight and the corresponding hopcount(s). Our results generalise earlier work of Bhamidi, van der Hofstad and Hooghiemstra, who assume that edge weights have an absolutely continuous distribution. The main tool we employ is the method of moments.Publicationtype: Journal Article16 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Dickman approximation of weighted sums of independent random variables in the Kolmogorov distanceWe consider distributional approximation by generalized Dickman distributions, which appear in number theory, perpetuities, logarithmic combinatorial structures and many other areas. We prove bounds in the Kolmogorov distance for the approximation of certain weighted sums of Bernoulli and Poisson random variables by members of this family.While such results have previously been shown in Bhattacharjee and Goldstein (2019) for distances based on smoother test functions and for a special case of the random variables considered in this paper, results in the Kolmogorov distance are new. We also establish optimality of our rates of convergence by deriving lower bounds. As a result, some interesting phase transitions emerge depending on the choice of the underlying parameters. The proofs of our results mainly rely on the use of Stein’s method. In particular, we study the solutions of the Stein equation corresponding to the test functions associated to the Kolmogorov distance, and establish their smoothness properties. As applications, we study the runtime of the Quickselect algorithm, an edge-length statistic of a long-range percolation model, and the weighted depth in randomly grown simple increasing trees.Publicationtype: Journal Article9 - Some of the metrics are blocked by yourconsent settings
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 , Randomized algorithms to generate hypergraphs with given degree sequencesThe question whether there exists a hypergraph whose degrees are equal to a given sequence of integers is a well-known reconstruction problem in graph theory, which is motivated by discrete tomography. In this paper we approach the problem by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.Publicationtype: Journal Article13 28 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Large components in the subcritical Norros-Reittu model(2025-01-11); The Norros-Reittu model is a random (multi-)graph with n vertices and i.i.d. weights assigned to them. The number of edges between any two vertices follows an independent Poisson distribution whose parameter is increasing in the weights of the two vertices. Choosing a suitable weight distribution leads to a power-law behaviour of the degree distribution as observed in many real-world complex networks. We study this model in the subcritical regime, i.e. in the absence of a giant component. For each component, we count all its vertices to determine the component sizes and show convergence of the corresponding point process of (rescaled) component sizes to a Poisson process. More generally, one can also count only specific vertices per component, like leaves. From this one can deduce asymptotic results on the size of the largest component or the maximal number of leaves in a single component. The results also apply to the Chung-Lu model and the generalised random graph.Publicationtype: Journal Article5 27 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Normal approximation of Kabanov–Skorohod Integrals on poisson spaces(2024-06); ; We consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process.Publicationtype: Journal ArticleScopus© Citations 1 27 63 - 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 Article11 54 - 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: without files , Rates of mulivariate normal approximation for statistics in geometric probability(2023-02); We employ stabilization methods and second order Poincaré inequalities to establish rates of multivariate normal convergence for a large class of vectors (Hs(1), . . ., Hs(m)), s ≥ 1, of statistics of marked Poisson processes on Rd, d ≥ 2, as the intensity parameter s tends to infinity. Our results are applicable whenever the functionals Hs(i), i ∈ 1, . . ., m, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the d2-, d3-, and dconvex-distances and are in general unimprovable. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are governed by the rate of convergence of s−1 Cov(Hs(i), Hs(j)), i, j ∈ 1, . . ., m, to the limiting covariance, shown to be at most of order s−1/d. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.Publicationtype: Journal Article23 - 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 126