A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity -- the strongest form of positive dependence -- and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fr\'echet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. This model turns out to be the joint distribution of samples with duplicate records. In this context, we provide an application of invariant correlation to the statistical inference in the presence of sample duplication. Finally, we show that 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.
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