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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 Article10 - 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: without files , Large degrees in scale-free inhomogeneous random graphs(2022-02); We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution.Publicationtype: Journal Article73 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Non-standard limits for a family of autoregressive stochastic sequencesWe examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.Publicationtype: Journal Article104 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , The random connection model and functions of edge-marked poisson processes: Second order properties and normal approximation(2021-02); ; The random connection model is a random graph whose vertices are given by the points of a Poisson process and whose edges are obtained by randomly connecting pairs of Poisson points in a position dependent but independent way. We study first and second order properties of the numbers of components isomorphic to given finite connected graphs. For increasing observation windows in an Euclidean setting we prove qualitative multivariate and quantitative univariate central limit theorems for these component counts as well as a qualitative central limit theorem for the total number of finite components. To this end we first derive general results for functions of edge marked Poisson processes, which we believe to be of independent interest.Publicationtype: Journal Article48 - 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 - Some of the metrics are blocked by yourconsent settings
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