Poisson approximation by coupling

TL;DR

This paper employs coupling techniques and elementary probability results to provide a more general Poisson approximation for binomial distributions, extending beyond traditional convergence methods.

Contribution

It introduces a coupling-based approach to Poisson approximation that is more general and elementary compared to classical convergence proofs.

Findings

Provides a coupling-based proof of Poisson approximation for binomial distributions.

Extends the approximation results beyond the classical limit theorems.

Uses only elementary probability results, simplifying the proof process.

Abstract

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$ goes to infinity and $np$ converges to some $\lambda$. In this note we use instead the coupling technique to show a much more general result. Moreover, we only use elementary results from probability.