Poisson Approximation

Abstract

In the past few years, mathematicians have developed a powerful technique known as the Chen-Stein method [2, 5] for approximating the distribution of a sum of weakly dependent Bernoulli random variables. In contrast to many asymptotic methods, this approximation carries with it explicit error bounds. Let X _α be a Bernoulli random variable with success probability p _α, where α be a Bernoulli random variable with success probability set I. It is natural to speculate that the sum S = Σ_α∈IX_α is approximately Poisson with mean λ = Σ_α∈Ip_α. The Chen-Stein method estimates the error in this approximation using the total variation distance between two integer-valued random variables Y and {tZ}. This distance is defined by

$$ \left\| {\mathcal{L}\left( Y \right) - \mathcal{L}\left( Z \right)} \right\| = {\mathbf{ }}\mathop {\sup }\limits_{A \subset \mathcal{N}} {\mathbf{ }}|\Pr \left( {Y \in A} \right) - \Pr \left( {Z \in A} \right)|, $$

where $\mathcal{L}$ denotes distribution, and $\mathcal{N}$ denotes the integers. Taking A = {0} in this definition yields the useful inequality

$$ \left| {\Pr \left( {Y = 0} \right) - \Pr \left( {Z = 0} \right)} \right| \leqslant \left\| {\mathcal{L}\left( Y \right) - \mathcal{L}\left( Z \right)} \right\|. $$