## The scaling limit of Poisson-driven order statistics with applications in geometric probability

DC FieldValueLanguage
dc.contributor.authorSchulte, Matthias
dc.contributor.authorThaele, Christoph
dc.date.accessioned2021-12-23T16:05:01Z-
dc.date.available2021-12-23T16:05:01Z-
dc.date.issued2012
dc.identifier.issn03044149
dc.identifier.urihttps://osnascholar.ub.uni-osnabrueck.de/handle/unios/6739-
dc.description.abstractLet eta(t) be a Poisson point process of intensity t >= 1 on some state space Y and let f be a non-negative symmetric function on Y-k for some k >= 1. Applying f to all k-tuples of distinct points of eta(t) generates a point process xi(t) on the positive real half-axis. The scaling limit of xi(t) as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m-th smallest point of xi(t) is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-Ito chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen-Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry. (C) 2012 Elsevier B.V. All rights reserved.
dc.language.isoen
dc.publisherELSEVIER SCIENCE BV
dc.relation.ispartofSTOCHASTIC PROCESSES AND THEIR APPLICATIONS
dc.subjectChen-Stein method
dc.subjectDISTANCES
dc.subjectExtreme values
dc.subjectFLATS
dc.subjectGeometric probability
dc.subjectIntegral geometry
dc.subjectLimit theorems
dc.subjectMalliavin calculus
dc.subjectMathematics
dc.subjectOrder statistics
dc.subjectPoisson flats
dc.subjectPoisson process approximation
dc.subjectPoisson space
dc.subjectRandom polytopes
dc.subjectScaling limit
dc.subjectStatistics & Probability
dc.subjectStochastic geometry
dc.subjectTRIANGLES
dc.subjectU-statistics
dc.subjectWiener-Ito chaos decomposition
dc.titleThe scaling limit of Poisson-driven order statistics with applications in geometric probability
dc.typejournal article
dc.identifier.doi10.1016/j.spa.2012.08.011
dc.identifier.isiISI:000310045300011
dc.description.volume122
dc.description.issue12
dc.description.startpage4096
dc.description.endpage4120
dc.identifier.eissn1879209X
dc.publisher.placePO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS
dcterms.isPartOf.abbreviationStoch. Process. Their Appl.
dcterms.oaStatusGreen Submitted, Bronze