Conditional uncorrelation equals independence

TL;DR

This paper proves that for real-valued random variables, independence is equivalent to conditional uncorrelation over interval-based conditioning, extending previous results to all types of variables without density assumptions.

Contribution

It generalizes the characterization of independence via conditional uncorrelation, removing the need for joint density and applying to all random variable types.

Findings

Independence equals conditional uncorrelation over intervals.

The characterization holds for all random variables, regardless of density.

Numerical examples demonstrate practical applicability.

Abstract

We show that the stochastic independence of real-valued random variables is equivalent to the conditional uncorrelation, where the conditioning takes place over the Cartesian products of intervals. Next, we express the mutual independence in terms of the conditional correlation matrix. Our results extend the results of Jaworski et al. (Electron. J. Stat., 18(1), 653-673, 2024), which are based on the copula functions and assume the existence of the joint density of the variables. We relax this assumption and show that the independence characterization via conditional uncorrelation is valid in full generality - that is, for all kinds of random variables and any dependencies between them. Additionally, we analyse the assumptions under which the independence is determined by the local uncorrelation. The measure-theoretic methodology we present uses the Radon-Nikodym derivative to reduce the multidimensional characterization problem to the simple one-dimensional conditioning. To demonstrate the potential usefulness of the presented results, various numerical examples are presented.