On Clustering of Random Points in the Plane and in Space

Abstract: If n points are independently and uniformly distributed in a large rectangular parallelepiped, A in R^d, 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 kn, if the sets in {C_n} decrease at a certain rate as n tends to infinity.

To each k-subset, which can be covered by some translate of C, can be attached its position on A, and the smallest s in R⁺ for which some translate of sC covers the k points. Poisson process approximation of three point processes determined by these positions and sizes are dealt with in the third paper.

All approximations are considered also when the total number of points is not fixed but have a Poisson distribution, so that the points constitute a Poisson process.