How to transform correlated random variables into uncorrelated ones

AbstractFor an arbitrary random vector X = (X1, X2,…, Xn), we can always construct uncorrelated random variables Y1, Y2,…, Yn and (R → R) functions f1, f2,…, fn, such that (X1, X2,…, Xn) = (f1(Y1), f2(Y2),…, fn(Yn)). Although the fs cannot always be one-to-one, in many important cases, the fs are not only one-to-one but also piecewise linear, e.g., if X is normally distributed. (This way, in many statistical models, the nuisance parameters can easily be transformed, such that their MLEs become uncorrelated with other parameters.