Conditional uncorrelation equals independence

TL;DR

This paper characterizes stochastic independence of real-valued random variables via conditional moments on rectangular event sets, showing that independence is equivalent to conditional uncorrelation on $U=\{X\in A, Y\in B\}$ with $A,B$ from intervals. It develops a measure-theoretic approach using the Radon-Nikodym derivative to reduce multidimensional conditioning to one-dimensional conditioning, thereby removing the joint-density requirement. A multivariate extension expresses independence through a conditional correlation matrix $\Sigma_U$, and the authors analyze when local uncorrelation suffices under appropriate support assumptions. The results provide a robust framework for independence testing based on conditional moments, with practical numerical demonstrations and guidance on estimation from finite samples.

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.

Figures (4)

• Figure 1: $200$ points are sampled from the probability distribution of the vector $(X,Y)$. The red dashed line is the contour of the square $U=[-1,1]^2$ and the green dashed line is the contour of the set $V=\{(x,y)\colon x\geq0,y\geq0\}$
• Figure 2: $180$ pairs of observations $X$ and $Y$ are sampled. The black line is the global line of linear regression between $Y$ and $X$, and the red continuous line is the line of linear regression between $Y$ and $X$ conditioning on the set determined by the quantile $q_Y=\frac{1}{3}$
• Figure 3: While the global empirical correlation between the residua and the fitted values equals zero, we observe the dependence on the left tail of the fitted values given by the quantile $q=0.2$
• Figure 4: While $X$ and $Y$ are visibly dependent, they are independent locally. The red lines are conditional regression lines and the black line is the global regression line, the sample size $n=10^3$.

Theorems & Definitions (23)

• Lemma 1
• proof
• Example 1
• Lemma 3
• proof
• Example 2
• Theorem 4
• proof
• Remark 1
• Theorem 5