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

Matthias Schulte and Christoph Thäle

Stochastic Processes and their Applications, 2012, vol. 122, issue 12, 4096-4120

Abstract: 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–Itô 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.

Keywords: Chen–Stein method; Extreme values; Geometric probability; Integral geometry; Limit theorems; Malliavin calculus; Order statistics; Poisson flats; Poisson process approximation; Poisson space; Random polytopes; Scaling limit; Stochastic geometry; U-statistics; Wiener–Itô chaos decomposition (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (4)

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DOI: 10.1016/j.spa.2012.08.011

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