# A note on invariance of the Cauchy and related distributions

## TL;DR

This paper explores the invariance properties of the Cauchy distribution under specific analytic transformations of the complex upper half-plane, providing insights into its stability under certain mappings.

## Contribution

It presents three new results related to the invariance of the Cauchy distribution under analytic self-maps of the upper half-plane that fix the point i.

## Key findings

- Identifies conditions under which the Cauchy distribution remains invariant.
- Provides new characterizations of analytic maps preserving the Cauchy distribution.
- Enhances understanding of distribution invariance in complex analysis.

## Abstract

It is known that if $f$ is an analytic self map of the complex upper half-plane which also maps $\mathbb{R}\cup\{\infty\}$ to itself, and $f(i)=i$, then $f$ preserves the Cauchy distribution. This note concerns three results related to the above fact.