Central limit theorems for motion-invariant Poisson hyperplanes in expanding convex bodies. (English) Zbl 1470.60049
Caristi, Giuseppe (ed.), Proceedings of the VIIth international conference in “Stochastic geometry, convex bodies, empirical measures and applications to mechanics and engineering train-transport”, Messina, Italy, April 22–24, 2009. Palermo: Circolo Matematico di Palermo. Suppl. Rend. Circ. Mat. Palermo (2) 81, 187-212 (2009).
Summary: It has been recently proved in the author et al. [Ann. Appl. Probab. 16, No. 2, 919–950 (2006; Zbl 1132.60023)] that the total number \(\Psi_0(B_\varrho^d)\) of intersection points generated by a stationary process of \(d\)-dimensional Poisson hyperplanes in a ball \(B_\varrho^d\) with radius \(\varrho\) is asymptotically normally distributed for large \(\varrho\). In the present paper we generalize this and other results by replacing \(B_\varrho^d\) by an expanding sampling window \(\varrho K\), where \(K\) is some fixed convex body with inner points. If the Poisson hyperplane process is additionally isotropic, the asymptotic variance of the scaled number \(\Psi_0(\varrho K)/\varrho^{d-1/2}\) of intersection points in \(\varrho K\) can be expressed in terms of a non-additive, motion-invariant ovoid functional of \(K\), which is calculated explicitly for the unit ball \(B_1^d\), ellipses and rectangles. Moreover, the method of \(U\)-statistics applied to the Poisson distributed number of hyperplanes hitting \(\varrho K\) allows to derive multivariate central limit theorems for the vector of numbers of intersection \(k\)-flats (\(k=0,1,\ldots,d-1\)) hitting \(\varrho K\) as well as for the vector of their total \(k\)-volumes (\(k=0,1,\ldots,d-1\)) within \(\varrho K\).
For the entire collection see [Zbl 1390.52002].
For the entire collection see [Zbl 1390.52002].
MSC:
60D05 | Geometric probability and stochastic geometry |
60F05 | Central limit and other weak theorems |
52A40 | Inequalities and extremum problems involving convexity in convex geometry |
53C65 | Integral geometry |