On Stein's method and mod-* convergence

TL;DR

This paper explores the connection between Stein's method and mod-* convergence, showing how the latter can be viewed as a higher order distributional approximation and providing a refined Berry-Esseen theorem for mod-Gaussian convergence.

Contribution

It establishes a link between Stein's method and mod-* convergence, offering a new perspective and refined results for distributional approximations.

Findings

Mod-* convergence can be interpreted as higher order distributional approximation.

A refined Berry-Esseen theorem is proved for sequences with mod-Gaussian convergence.

The connection enhances understanding of distributional limits beyond traditional metrics.

Abstract

Stein's method allows to prove distributional convergence of a sequence of random variables and to quantify it with respect to a given metric such as Kolmogorov's (a Berry-Ess\'een type theorem). Mod-* convergence quantifies the convergence of a sequence of random variables to a given distribution in a sense unusual in probability theory, a priori unrelated to a metric on probability measures. This article gives a connection between these two notions. It shows that mod-* convergence can be understood as a higher order approximation in distribution when the limiting function is integrable and proves a refined Berry-Ess\'een type theorem for sequences converging in the mod-Gaussian sense.