On Clustering of Random Points in the Plane and in Space

TL;DRAbstract

If n points are independently and uniformly distributed in a large rectangular parallelepiped, A in R<sup>d</sup>, 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.<p /> In the latter part of the first paper the number of subsets consisting of k<n points which can be covered by some translate of C in A is considered in the plane. When n is large and C is small, this number is approximately Poisson, as shown by means of the Stein-Chen method. This approximation is used in some

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