Rare events and Poisson point processes

TL;DR

This paper interprets earlier results on approximating sums of independent terms by compound Poisson laws as precise estimates for how closely a sample of rare independent events resembles a Poisson point process after Poissonization.

Contribution

It provides a new interpretation of existing approximation results as sharp quantitative estimates for the convergence of rare event samples to Poisson point processes.

Findings

Quantitative bounds for the approximation of rare event samples by Poisson processes

Interpretation of compound Poisson laws as approximations for rare event distributions

Enhanced understanding of Poissonization in the context of rare events

Abstract

The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent terms by the accompanying compound Poisson laws may be interpreted as rather sharp quantitative estimates for the closeness between the sample containing independent observations of rare events and the Poisson point process which is obtained after a Poissonization of the initial sample.