The Number of Points of an Empirical or Poisson Process Covered by Unions of Sets

✍ Scribed by P. Auer; K. Hornik

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 602 KB
• Volume: 57
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.0020

✦ Synopsis

For the d-dimensional empirical process and the d-dimensional homogeneous Poisson process we give weak laws for the maximal and minimal number of points in unions of sets from certain families.

1996 Academic Press, Inc.

1. Introduction and Results

In Auer and Hornik [1] we investigated the maximal and minimal number of points of a Poisson process in single sets from certain families. In this paper we obtain some surprising results containing the number of points in unions of sets from such families. Furthermore, we investigate the number of points of an empirical process in such unions of sets.

Let % n be the empirical process on [0, 1] d generated by n points from the uniform distribution on [0, 1] d , let ' be a homogeneous Poisson process on R d with parameter *, and let E be some fixed family of Borel measurable subsets of [0, 1] d . We are, for positive integer L, interested in the quantities article no.