A note on the equivalence between the conditional uncorrelation and the independence of random variables
Statistics Theory
2024-04-22 v1 Probability
Statistics Theory
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.
Keywords
Cite
@article{arxiv.2210.16655,
title = {A note on the equivalence between the conditional uncorrelation and the independence of random variables},
author = {Piotr Jaworski and Damian Jelito and Marcin Pitera},
journal= {arXiv preprint arXiv:2210.16655},
year = {2024}
}