The distance problem via subadditivity

TL;DR

This paper characterizes which distributions can be represented as the distance between two independent random variables in a metric space, showing that all nonnegative discrete distributions with support including zero, and some finitely supported distributions, can be expressed this way.

Contribution

It provides a complete characterization of distributions that can be realized as distances between independent random variables in a metric space.

Findings

All nonnegative discrete distributions with support containing zero can be represented as such distances.

A class of finitely supported distributions with density can also be expressed as distances.

The results answer a question posed by Aldous, Blanc, and Curien.

Abstract

In a recent paper, Aldous, Blanc and Curien asked which distributions can be expressed as the distance between two independent random variables on some separable measured metric space. We show that every nonnegative discrete distribution whose support contains $0$ arises in this way, as well as a class of finitely supported distributions with density.