On the number of points of a homogeneous Poisson process
type: Item
instance of: scholarly article
title: On the number of points of a homogeneous Poisson process
zbMATH Open document ID: 0797.60029
DOI: 10.1016/0047-259X(94)80008-J
author: Peter Auer
author: Kurt Hornik
published in: Journal of Multivariate Analysis
publication date: 1994-10-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 566960
zbMATH Keywords: strong laws
zbMATH Keywords: homogeneous \(d\)-dimensional Poisson process
zbMATH Keywords: maximal and minimal number of points
zbMATH Keywords: multivariate spacings
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 566960