Distances Between Poisson k-Flats
|Central limit theorem; Chaos decomposition; Extreme values; LIMIT; Limit theorems; Mathematics; Poisson flat process; Poisson point process; Poisson U-statistic; Statistics & Probability; Stochastic geometry; Wiener-Ito integral
|METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY
The distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k < d/2 are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the m-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real half-axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener-It chaos decomposition and the Malliavin-Stein method.
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