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 multiple examples.
Funding Statement
Marcin Pitera acknowledges support from the National Science Centre, Poland, via project 2020/37/B/HS4/00120. Part of the work of Damian Jelito was funded by the Priority Research Area Digiworld under the program Excellence Initiative – Research University at the Jagiellonian University in Kraków.
Citation
Piotr Jaworski. Damian Jelito. Marcin Pitera. "A note on the equivalence between the conditional uncorrelation and the independence of random variables." Electron. J. Statist. 18 (1) 653 - 673, 2024. https://doi.org/10.1214/24-EJS2212
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