Exponential Approximation by Stein's Method and Spectral Graph Theory

TL;DR

This paper develops Berry-Esseen bounds for the exponential distribution using Stein's method and applies it to analyze the spectral distribution of a specific Markov chain, marking a novel use of Stein's method in spectral graph theory.

Contribution

It introduces the first application of Stein's method to study the spectrum of a graph with a non-normal limit, providing sharp error bounds.

Findings

Established Berry-Esseen bounds for exponential approximation.

Obtained a sharp error term for the spectrum of the Bernoulli-Laplace Markov chain.

Demonstrated the first use of Stein's method in spectral graph analysis with non-normal limits.

Abstract

General Berry-Esseen bounds are developed for the exponential distribution using Stein's method. As an application, a sharp error term is obtained for Hora's result that the spectrum of the Bernoulli-Laplace Markov chain has an exponential limit. This is the first use of Stein's method to study the spectrum of a graph with a non-normal limit.