Poisson Approximation to the Convolution of Power Series Distributions

TL;DR

This paper develops bounds on the total variance distance between Poisson and convolution of power series distributions using Stein's method, unifying various discrete distribution approximations and comparing them with negative binomial results.

Contribution

It introduces a unified approach to Poisson approximation for power series distributions and derives new error bounds using Stein's method.

Findings

Provides explicit error bounds for Poisson approximation

Unifies approximation results for many discrete distributions

Compares Poisson and negative binomial approximations for specific sums

Abstract

In this article, we obtain, for the total variance distance, the error bounds between Poisson and convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several Poisson limit theorems follow as corollaries from our bounds. As applications, we compare the Poisson approximation results with the negative binomial approximation results, for the sums of Bernoulli, geometric, and logarithmic series random variables.