Invariant correlation under marginal transforms

The Pearson correlation coefficient is generally not invariant under common marginal transforms, but such an invariance property may hold true for specific models such as independence. A bivariate random vector is said to have an invariant correlation if its Pearson correlation coefficient remains unchanged under any common marginal transforms. We characterize all models of such a random vector via a certain combination of independence and the strongest positive dependence called comonotonicity. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fr\'echet copulas. We then extend the concept of invariant correlation to multi-dimensional models, and characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model which admits any prescribed invariant correlation matrix. Finally, all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.