Stein's method and Poisson process approximation
2005; Linguagem: Inglês
10.1142/9789812567680_0003
ISSN1793-0758
Autores Tópico(s)Stochastic processes and statistical mechanics
ResumoLecture Notes Series, Institute for Mathematical Sciences, National University of SingaporeAn Introduction to Stein's Method, pp. 115-181 (2005) No AccessStein's method and Poisson process approximationAihua XiaAihua XiaDepartment of Mathematics and Statistics, University of Melbourne, VIC 3010, AustraliaDepartment of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singaporehttps://doi.org/10.1142/9789812567680_0003Cited by:11 (Source: Crossref) PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The chapter begins with an introduction to Poisson processes on the real line and then to Poisson point processes on a locally compact complete separable metric space. The focus is on the characterization of Poisson point processes. The next section reviews the basics of Markov immigration-death processes, and then of Markov immigration-death point processes. We explain how a Markov immigration-death point process evolves, establish its generator and find its equilibrium distribution. We then discuss our choice of metrics for Poisson process approximation, and illustrate their use in the context of stochastic calculus bounds for the accuracy of Poisson process approximation to a point process on the real line. These considerations are combined in constructing Stein's method for Poisson process approximation. Some of the key estimates given here are sharper than those found elsewhere in the literature, and have simpler proofs. In the final part, we show how to apply the bounds in various examples, from the easiest Bernoulli process to more complicated networks of queues. FiguresReferencesRelatedDetailsCited By 11Cited by lists all citing articles based on Crossref citation.Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision treesGünter Last, Giovanni Peccati and D. Yogeshwaran15 March 2023 | Random Structures & Algorithms, Vol. 63, No. 2Poisson process approximation under stabilization and Palm couplingOmer Bobrowski, Matthias Schulte and D. Yogeshwaran16 December 2022 | Annales Henri Lebesgue, Vol. 5Poisson approximationS. Y. Novak1 Jan 2019 | Probability Surveys, Vol. 16, No. noneBounds for the Probability Generating Functional of a Gibbs Point ProcessKaspar Stucki and Dominic Schuhmacher22 February 2016 | Advances in Applied Probability, Vol. 46, No. 1Approximating dependent rare eventsLouis H. Y. Chen and Adrian Röllin1 Sep 2013 | Bernoulli, Vol. 19, No. 4References7 February 2012Poisson process approximation for dependent superposition of point processesLouis H.Y. Chen and Aihua Xia1 May 2011 | Bernoulli, Vol. 17, No. 2Spatial logistic regression and change-of-support in Poisson point processesA. Baddeley, M. Berman, N.I. Fisher, A. Hardegen and R.K. Milne et al.1 Jan 2010 | Electronic Journal of Statistics, Vol. 4, No. noneDistance estimates for dependent thinnings of point processes with densitiesDominic Schuhmacher1 Jan 2009 | Electronic Journal of Probability, Vol. 14, No. noneA new metric between distributions of point processesDominic Schuhmacher and Aihua Xia1 July 2016 | Advances in Applied Probability, Vol. 40, No. 3Upper Bounds for Stein-Type OperatorsFraser Daly1 Jan 2008 | Electronic Journal of Probability, Vol. 13, No. none Recommended An Introduction to Stein's MethodMetrics History PDF download
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