Page 310 - Applied Probability

14
Poisson Approximation
14.1 Introduction

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 ranges over some finite index set I. It is natural to speculate that

the sum S = X α is approximately Poisson with mean λ = p α .
α∈I α∈I
The Chen-Stein method estimates the error in this approximation using
the total variation distance between two integer-valued random variables
Y and Z. This distance is defined by
L(Y ) −L(Z) = sup | Pr(Y ∈ A) − Pr(Z ∈ A)|,
A⊂N
where L denotes distribution, and N denotes the integers. Taking A = {0}
in this definition yields the useful inequality

| Pr(Y =0) − Pr(Z =0)|≤ L(Y ) −L(Z) .
The coupling method is one technique for explicitly bounding the total

variation distance between S = X α and a Poisson random variable
α∈I
Z with the same mean λ [5, 15]. In many concrete examples, it is possible
to construct for each α two random variables U α and V α on a common
probability space in such a way that V α is distributed as S − 1 conditional
on the event X α = 1 and U α is distributed as S unconditionally. The bound
1 − e −λ
L(S) −L(Z) Ø p α E(|U α − V α |) (14.1)
λ
α∈I
then applies. Because U α and V α live on the same probability space, they
are said to be coupled. If U α ≥ V α for all α, then the simplified bound

1 − e −λ
L(S) −L(Z) Ø [λ − Var(S)] (14.2)
λ
holds. Inequality (14.2) shows that Var(S) ≈ E(S) is a sufficient as well as
a necessary condition for S to be approximately Poisson.
The neighborhood method of bounding the total variation distance
exploits certain neighborhoods of dependency B α associated with each α