Conditional uncorrelation equals independence

Abstract: We express the independence of real-valued random variables in terms of 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. While the previous studies on the subject are based on the copula functions, our approach uses the Radon-Nikodym derivative to reduce the general problem to the simple one-dimensional conditioning.