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Rates of mulivariate normal approximation for statistics in geometric probability
Publikationstyp
Journal Article
Date Issued
2023-02
Sprache
English
Author(s)
Institut
Volume
33
Issue
1
Start Page
507
End Page
548
Citation
Annals of Applied Probability 33 (1): 507-548 (2023-02-01)
Publisher DOI
Scopus ID
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.
Subjects
Multivariate normal approximation
multivariate statistics in geometric probability
random Euclidean graphs
stabilization
stochastic geometry
DDC Class
510: Mathematik