Approximating dependent rare events

TL;DR

This paper reviews the development of Poisson approximation via Stein's method, highlights recent applications in arithmetic progressions and bootstrap percolation, and discusses extensions and open problems in the field.

Contribution

It provides a comprehensive historical overview, presents new applications, and discusses generalizations of Poisson approximation methods.

Findings

Poisson approximation effectively models dependent rare events

Recent applications include maximal arithmetic progressions and bootstrap percolation

Open problems in multivariate and compound Poisson approximation are identified

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.