Bounding Kolmogorov distances through Wasserstein and related integral probability metrics

TL;DR

This paper derives general upper bounds on the Kolmogorov distance between probability distributions using Wasserstein metrics, extending previous results and applying them to various distribution approximations in Stein's method.

Contribution

It introduces new bounds linking Kolmogorov and Wasserstein distances, broadening the applicability of existing approximation techniques.

Findings

Bounds for Kolmogorov distance in terms of Wasserstein metrics

Application to multivariate normal, beta, and variance-gamma approximations

Generalization of existing results in probability metric bounds

Abstract

We establish general upper bounds on the Kolmogorov distance between two probability distributions in terms of the distance between these distributions as measured with respect to the Wasserstein or smooth Wasserstein metrics. These bounds generalise existing results from the literature. To illustrate the broad applicability of our general bounds, we apply them to extract Kolmogorov distance bounds from multivariate normal, beta and variance-gamma approximations that have been established in the Stein's method literature.