Approximating dependent rare events

Abstract

In this paper we give a historical account of the development of Poisson approximation using Stein's method and present some of the main results. We give two recent applications, one on maximal arithmetic progressions and the other on bootstrap percolation. We also discuss generalisations to compound Poisson approximation, Poisson process approximation and multivariate Poisson approximation, and state a few open problems.

Published in at http://dx.doi.org/10.3150/12-BEJSP18 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (4)

• Exchangeable pairs and Poisson approximation
• Degree asymptotics with rates for preferential attachment random graphs
• The Chen-Stein method for Poisson functionals
• On the longest length of arithmetic progressions

Cited by in corpus (6)

• Wilson loop expectations in lattice gauge theories with finite gauge groups
• On Stein operators for discrete approximations
• Stein meets Malliavin in normal approximation
• Tractably Modelling Dependence in Networks Beyond Exchangeability
• Wilson Loop Expectations for Non-Abelian Gauge Fields Coupled to a Higgs Boson at Low and High Disorder
• Compound Poisson approximations in ${\ell }_p$-norm for sums of weakly dependent vectors