Stein's method and the Laplace distribution

TL;DR

This paper develops Stein's method techniques to bound approximation errors for the Laplace distribution, applying it to geometric sums and deriving a Berry-Esseen type theorem for convergence of random sums.

Contribution

It introduces a new framework using second order characterizing equations and distributional transformations related to zero-biasing for Laplace approximation.

Findings

Bounded error terms for Laplace approximation using Stein's method

Established a Berry-Esseen type theorem for geometric sum convergence

Connected zero-biasing techniques to Laplace distribution approximation

Abstract

Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing.