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.

翻译：皮尔逊相关系数通常不适用于常见的边际变换，但这种不变性性质在特定模型（如独立模型）中可能成立。若二元随机向量在任意公共边际变换下其皮尔逊相关系数保持不变，则称其具有不变相关性。我们通过独立性与最强正相关结构（即共同单调性）的特定组合，刻画了此类随机向量的所有模型。特别地，我们证明具有不变相关性的可交换copula类恰好可由所谓的正弗雷歇copula精确描述。进而将不变相关性概念推广至多维模型，并通过团划分多面体刻画所有不变相关性矩阵的集合。我们还提出了一个正回归相依模型，该模型可容纳任意指定的不变相关性矩阵。最后，除一个特殊情形外，若将公共边际变换限制为单调递增变换，所有不变相关性的刻画结论均保持不变。