A characterization of a Cauchy family on the complex space

TL;DR

This paper characterizes a unique family of distributions on the complex space based on invariance under a specific group of transformations, and compares it with existing families.

Contribution

It introduces a novel characterization of a Cauchy family on the complex space using group invariance properties.

Findings

The family is uniquely determined by its invariance under a certain transformation group.

Comparison shows how this family relates to other known distribution families.

The characterization provides new insights into the structure of distributions on complex spaces.

Abstract

It is shown that a family of distributions on the complex space is characterized as the only family such that the orbit of one distribution under a certain group of transformations on the complex space is the same as that under the group of affine transformations. The resulting family is compared with some existing families.