TUHH Open Research (TORE)https://tore.tuhh.deTORE captures, stores, indexes, preserves, and distributes digital research material.Mon, 05 Dec 2022 20:39:55 GMT2022-12-05T20:39:55Z5071Non-standard limits for a family of autoregressive stochastic sequenceshttp://hdl.handle.net/11420/10471Title: Non-standard limits for a family of autoregressive stochastic sequences
Authors: Foss, Sergey; Schulte, Matthias
Abstract: We 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.
Mon, 20 Sep 2021 00:00:00 GMThttp://hdl.handle.net/11420/104712021-09-20T00:00:00ZTesting multivariate uniformity based on random geometric graphshttp://hdl.handle.net/11420/8302Title: Testing multivariate uniformity based on random geometric graphs
Authors: Ebner, Bruno; Nestmann, Franz; Schulte, Matthias
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11420/83022020-01-01T00:00:00ZPoisson approximation with applications to stochastic geometryhttp://hdl.handle.net/11420/11357Title: Poisson approximation with applications to stochastic geometry
Authors: Pianoforte, Federico; Schulte, Matthias
Abstract: This 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.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/11420/113572021-01-01T00:00:00ZThe random connection model and functions of edge-marked poisson processes: Second order properties and normal approximationhttp://hdl.handle.net/11420/11811Title: The random connection model and functions of edge-marked poisson processes: Second order properties and normal approximation
Authors: Last, Günter; Nestmann, Franz; Schulte, Matthias
Abstract: 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.
Mon, 01 Feb 2021 00:00:00 GMThttp://hdl.handle.net/11420/118112021-02-01T00:00:00ZCriteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellationshttp://hdl.handle.net/11420/11789Title: Criteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellations
Authors: Pianoforte, Federico; Schulte, Matthias
Abstract: This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables, point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson–Voronoi tessellations and large nearest neighbor distances in a Boolean model of disks.
Mon, 07 Feb 2022 00:00:00 GMThttp://hdl.handle.net/11420/117892022-02-07T00:00:00ZVariance asymptotics and central limit theory for geometric functionals of Poisson cylinder processeshttp://hdl.handle.net/11420/13380Title: Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes
Authors: Betken, Carina; Schulte, Matthias; Thäle, Christoph
Abstract: 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.
Wed, 01 Jun 2022 00:00:00 GMThttp://hdl.handle.net/11420/133802022-06-01T00:00:00ZLarge degrees in scale-free inhomogeneous random graphshttp://hdl.handle.net/11420/12119Title: Large degrees in scale-free inhomogeneous random graphs
Authors: Bhattacharjee, Chinmoy; Schulte, Matthias
Abstract: 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.
Tue, 01 Feb 2022 00:00:00 GMThttp://hdl.handle.net/11420/121192022-02-01T00:00:00Z