On Clustering of Random Points in the Plane and in Space

Doctoral thesis, 1996

If n points are independently and uniformly distributed in a large rectangular parallelepiped, A in Rd, d =2,3, they may possibly aggregate in such a way that they are contained in some translate of a given convex set C in A. If the points are replaced by copies of C, these translated sets may have a non-empty intersection. The probabilities of these two events are in fact equal. This thesis consists of three separate papers, of which parts of the first and the second are devoted to the derivation of this probability. Also generalizations are considered, which allow the sets to be unequal, and to be rotated according to a uniform distribution. In the latter part of the first paper the number of subsets consisting of k

scan statistics

geometric probability

integral geometry

Poisson approximation

mixed volumes

Stein-Chen method

random intersections

convex sets

convergence of point processes