Stein's Method for Probability Distributions on $\mathbb{S}^1$

TL;DR

This paper adapts Stein's method to the geometry of the unit circle, providing bounds on distances between circular distributions like von-Mises, wrapped normal, and wrapped Cauchy.

Contribution

It introduces a modified density approach for Stein's method tailored to the circle's geometry, enabling new bounds between circular distributions.

Findings

Derived an upper bound for the Wasserstein metric on $\

Established bounds between key circular distributions such as von-Mises and wrapped normal.

Abstract

In this paper, we propose a modification to the density approach to Stein's method for intervals for the unit circle $\mathbb{S}^1$ which is motivated by the differing geometry of $\mathbb{S}^1$ to Euclidean space. We provide an upper bound to the Wasserstein metric for circular distributions and exhibit a variety of different bounds between distributions; particularly, between the von-Mises and wrapped normal distributions, and the wrapped normal and wrapped Cauchy distributions.