Self-Inverse and Exchangeable Random Variables

TL;DR

This paper characterizes self-inverse random variables as ratios of exchangeable variables, providing a theoretical link between reciprocal symmetry and exchangeability in probability distributions.

Contribution

It establishes that a ratio of two variables is self-inverse if and only if the variables are exchangeable, offering a new perspective on reciprocal symmetry.

Findings

Self-inverse variables are characterized as ratios of exchangeable variables.

Exchangeability is necessary and sufficient for a ratio to be self-inverse.

Not all self-inverse variables can be represented as ratios of iid variables.

Abstract

A random variable Z will be called self-inverse if it has the same distribution as its reciprocal 1/Z. It is shown that if Z is defined as a ratio, X/Y, of two rv's X and Y (with Pr[X=0]=Pr[Y=0]=0), then Z is self-inverse if and only if X and Y are (or can be chosen to be) exchangeable. In general, however, there may not exist iid X and Y in the ratio representation of Z.