Abstract
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–Itô chaos decomposition and the Malliavin–Stein method.
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Schulte, M., Thäle, C. Distances Between Poisson k -Flats . Methodol Comput Appl Probab 16, 311–329 (2014). https://doi.org/10.1007/s11009-012-9319-2
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DOI: https://doi.org/10.1007/s11009-012-9319-2
Keywords
- Central limit theorem
- Chaos decomposition
- Extreme values
- Limit theorems
- Poisson flat process
- Poisson point process
- Poisson U-statistic
- Stochastic geometry
- Wiener–Itô integral