On existence and uniqueness of univariate Stein kernels

TL;DR

This paper characterizes univariate distributions that admit Stein kernels, providing a complete classification and applying the results to establish an optimal rate in the central limit theorem for these distributions.

Contribution

It offers a complete characterization of distributions with Stein kernels and applies this to prove a sharp quantitative CLT result.

Findings

Complete characterization of distributions with Stein kernels

Concrete examples of distributions admitting Stein kernels

Optimal $n^{-1/2}$ rate in total variation for CLT within this class

Abstract

We completely characterize the class of univariate distributions allowing for a Stein kernel and illustrate our result by means of some concrete distributions. Moreover, we apply our findings to prove a quantitative version of the central limit theorem with optimal rate $n^{-1/2}$ in total variation distance for i.i.d. random variables whose distribution belongs to that class.