Point processes on spaces of flats and other homogeneous spaces

Abstract

The present paper is primarily concerned with the structure of certain random sets of flats in n-dimensional Euclidean space Rn. To be more specific, let be the space of k-dimensional affine subspaces of Rn (to be called k-flats below), equipped with its natural topology ((14), p. 132). is a topological homogeneous space under the action of the group of Euclidean displacements in Rn and our purpose is to give conditions under which a stochastic point process on , stationary under , is a doubly stochastic Poisson process. The results will be seen to apply equally well to abstract spaces sharing some of the structural properties of .