  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.
