A Simple Necessary Condition For Independence of Real-Valued Random Variables

TL;DR

This paper introduces a simple necessary condition based on support sets to determine the dependence of two real-valued random variables, simplifying the process without requiring full distribution factorization.

Contribution

It presents a novel support-based necessary condition for independence, applicable to both discrete and continuous real-valued random variables, with proofs and illustrative examples.

Findings

The condition simplifies independence testing by examining support sets.

It applies to discrete, continuous, and general real-valued variables.

Illustrated through three simple examples.

Abstract

The standard method to check for the independence of two real-valued random variables -- demonstrating that the bivariate joint distribution factors into the product of its marginals -- is both necessary and sufficient. Here we present a simple necessary condition based on the support sets of the random variables, which -- if not satisfied -- avoids the need to extract the marginals from the joint in demonstrating dependence. We review, in an accessible manner, the measure-theoretic, topological, and probabilistic details necessary to establish the background for the old and new ideas presented here. We prove our result in both the discrete case (where the basic ideas emerge in a simple setting), the continuous case (where serious complications emerge), and for general real-valued random variables, and we illustrate the use of our condition in three simple examples.