Mod-Poisson approximation schemes and higher-order Chen-Stein inequalities

TL;DR

This paper extends the Chen-Stein inequality for Poisson approximation, improving convergence rates and accommodating dependent variables using mod-Poisson convergence and combinatorial methods, applicable to various probabilistic models.

Contribution

It introduces a unified framework for Poisson approximation that enhances convergence rates and handles dependencies via mod-Poisson convergence and elementary symmetric functions.

Findings

Improved convergence rates using signed or positive measures.

Extension to dependent random variables.

Applicability to number theory and permutation models.

Abstract

In this article, we provide an extension of the Chen-Stein inequality for Poisson approximation in the total variation distance for sums of independent Bernoulli random variables in two ways. We prove that we can improve the rate of convergence (hence the quality of the approximation) by using explicitly constructed signed or positive probability measures, and that we can extend the setting to possibly dependent random variables. The framework which allows this is that of mod-Poisson convergence, and more precisely those mod-Poisson convergent sequences whose residue functions can be expressed as a specialization of the generating series of elementary symmetric functions. This combinatorial reformulation allows us to have a general and unified framework in which we can fit the classical setting of sums of independent Bernoulli random variables as well as other examples coming e.g. from probabilistic number theory and random permutations.