The scaling limit of Poisson-driven order statistics with applications in geometric probability

By Matthias Schulte, C. Thaele. 2012

54 citations recorded by the source corpus.

Let ηt be a Poisson point process of intensity t≥1 on some state space Y and let f be a non-negative symmetric function on Yk for some k≥1. Applying f to all k-tuples of distinct points of ηt generates a point process ξt on the positive real half-axis. The scaling limit of ξt as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m-th smallest point of ξt is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Ito chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.

Subfields: Applied Mathematics, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/83d47cfc93e5956cb1ce8705b7efb6e35224ce2b

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Citations in the corpus

Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry (2014, methodology): “We remark that this form of M was tailored to the applications in [43]; a more general version is stated without proof in [44].”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “Other relevant contributions in this area are: [16, 49] for explicit bounds in the Kolmogorov distance; [18] for applications to additive functionals of Boolean models; [7, 14, 23, 24, 28] for applications to geometric U -statistics and other non-linear functionals; [50] for CLTs in the framework of”
Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs (2011, background): “Among other features, these operators provide a neat way to deal with the product of multiple stochastic integral, and virtually replace the use of diagram formulae – see e.g. [24].”
Central limit theorems for $U$-statistics of Poisson point processes (2011, background): “In their recent paper [34] this approach is further refined to obtain point process convergence for the intrinsic volumes of the intersection process of Poisson k-flats in the unit ball.”
New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, methodology): “Applications of findings dealing with the Gaussian case have found notable applications in the theory and statistics of Gaussian random processes [3, 4] (most prominently the fractional Brownian motion [16]), spherical random fields [1, 15], random matrix theory [18] and universality questions [20],”