On the number of points of a homogeneous Poisson process
Consider a homogeneous \(d\)-dimensional Poisson process. The authors study the limiting behaviour of maximal and minimal number of points in certain families of sets of volume \(V_ T\) in \([0,T]^ d\), where \(0< V_ T\leq T^ d\) and \(\lim_{T\to\infty} V_ T/T^ d= 0\). Weak (i.e. in probability) limit theorems can be established provided that one knows how ``rich'' the considered classes of sets are. To obtain strong (i.e. with probability one) results, more structure is needed concerning certain ``covering'' numbers of the classes involved. The latter have already been introduced by \textit{P. Deheuvels}, \textit{J. H. J. Einmahl}, \textit{D. M. Mason} and \textit{F. H. Ruymgaart} [ibid. 24, No. 1, 155-176 (1988; Zbl 0646.60035)], where corresponding results have been established for multivariate spacings.
homogeneous \(d\)-dimensional Poisson process
scientific article; zbMATH DE number 566960