Characterizations of the Cauchy distribution associated with integral transforms

TL;DR

This paper introduces two new simple characterizations of the Cauchy distribution using Möbius and Mellin transforms, also extending to circular and mixture Cauchy distributions.

Contribution

It provides novel characterization methods for the Cauchy distribution and related models via integral transforms, expanding understanding of their properties.

Findings

Characterizations of the Cauchy distribution using Möbius and Mellin transforms

Extensions to circular Cauchy distribution

Extensions to mixture Cauchy models

Abstract

We give two new simple characterizations of the Cauchy distribution by using the M\"obius and Mellin transforms. They also yield characterizations of the circular Cauchy distribution and the mixture Cauchy model.