On bounds in Poisson approximation for distributions of independent negative-binomial distributed random variables
Abstract
Using the Stein–Chen method some upper bounds in Poisson approximation for distributions of row-wise triangular arrays of independent negative-binomial distributed random variables are established in this note.
A Note on Poisson Approximation for Independent Geometric Random Variables
Let W be a sum of n independent geometric random variables. In 2007, Teerapabolarn and Wongkasem [4] used the Stein-Chen method to give a non-uniform bound in approximating the distribution function of W by the Poisson distribution function with mean λ = E(W )= � n i=1 qip −1 i , where qi =1 −pi. In this paper, a non-uniform bound on the such approximation has been given for the Poisson mean λ = � n=1 qi.
Coupling methods in approximations
Let X and Y be two arbitrary k-dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P(X ≠ Y). This result is used to find the least upper bound for the metric d(X, Y) = supA|P(X ∈ A) − P(Y ∈ A)| and to derive the inequality d(ΣXi, ΣYi) ≤ Σd(Xi, Yi). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given. 
 
 
 
Soient X et Y deux vecteurs aleatoires a k dimensions prenant leurs valeurs dans un espace discret. Nous etablissons l'existence d'une methode de couplage qui minimise P(X ≠ Y) ∈ A. Ce resultat permet ensuite de calculer le minorant de la metrique d(X, Y) = sup|(X ∈ A) - P(Y ∈ A)| et de de-montrer l'inegalite d(ΣXi, ΣYi) ≤ Σd(Xi, Yi). Ceci nous conduit a degager une methode integree permettant de mesurer l'ecart entre les distributions de sommes de vecteurs aleatoires independants.