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].