A note on the equivalence between the conditional uncorrelation and the independence of random variables

TL;DR

This paper demonstrates that localised conditional uncorrelation across all quantile sets implies independence of random variables, providing a new perspective on the relationship between correlation and independence.

Contribution

It establishes that localised conditional uncorrelation for all quantile sets is equivalent to independence, a reversal of the usual implication.

Findings

Conditional uncorrelation across all quantile sets implies independence.

Illustrated potential usefulness with two simple examples.

Focuses on the absolutely continuous case.

Abstract

It is well known that while the independence of random variables implies zero correlation, the opposite is not true. Namely, uncorrelated random variables are not necessarily independent. In this note we show that the implication could be reversed if we consider the localised version of the correlation coefficient. More specifically, we show that if random variables are conditionally (locally) uncorrelated for any quantile conditioning sets, then they are independent. For simplicity, we focus on the absolutely continuous case. Also, we illustrate potential usefulness of the stated result using two simple examples.