On Stein factors in Stein's method for normal approximation

TL;DR

This paper thoroughly characterizes the optimal uniform bounds for derivatives of solutions to the standard normal Stein equation, focusing on test functions with Lipschitz continuous derivatives, advancing the understanding of Stein's method for normal approximation.

Contribution

It provides a complete description of the best possible bounds for derivatives of Stein equation solutions under Lipschitz conditions, enhancing theoretical insights in Stein's method.

Findings

Identifies optimal uniform bounds for derivatives of Stein solutions.

Clarifies regularity conditions for test functions in Stein's method.

Advances theoretical understanding of normal approximation techniques.

Abstract

Building on the rather large literature concerning the regularity of the solution of the standard normal Stein equation, we provide a complete description of the best possible uniform bounds for the derivatives of the solution of the standard normal Stein equation when the test functions belong to the class of real-valued functions whose $k$-th order derivative is Lipschitz.