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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