Symmetric representations of distributions over $\mathbb{R}^2$ by distributions with not more than three-point supports

TL;DR

This paper presents a method to represent any two-dimensional distribution with a given mean as a convex combination of simpler distributions supported on at most three points, maintaining the same mean.

Contribution

It introduces a symmetric representation technique for 2D distributions using convex combinations of three-point supported distributions with identical means.

Findings

Any 2D distribution can be expressed as a convex combination of three-point supported distributions.

The method preserves the mean values of the original distribution.

The representation is symmetric and supports distributions with minimal complexity.

Abstract

We construct symmetric representations of distributions over two-dimensional plane with given mean values as convex combinations of distributions with supports containing not more than three points and with the same mean values.