On the existence of mosaic and indicator random fields with spherical, circular, and triangular variograms (Q616043)

The geometric covariogram of a set \(A\) is the function \(K(h)\) equal to the Lebesgue measure of the intersection of \(A\) with \(A-h\). Three important covariogram functions are obtained by taking \(A\) to be the unit ball in spaces of dimension 1, 2 or 3 and are called, respectively, triangular, circular and spherical covariograms. These covariograms are isotropic, depend only on \(\|h\|\) and so can be then considered in spaces of arbitrary dimension. The geometric covariogram of a random closed set \(X\) is defined as the expected Lebesgue measure of \(X\cap(X-h)\). The author investigates if the three basic covariograms (triangular, circular and spherical) may be realised as covariograms of several important random sets: the zero cell of a random tessellation (and so related to the variogram of the mosaic random field obtained by assigning i.i.d. values to individual cells), the Boolean random set, and the excursion set of a Gaussian random field. While it is shown that random sets on the line may have these covariograms, this is generally not possible in spaces of dimensions 2 and 3.