Bivariate Distributions with Given Marginals

Bivariate distributions with minimum and maximum correlations for given marginal distributions are characterized. Such extremal distributions were first introduced by Hoeffding (1940) and Frechet (1951). Several proofs are outlined including ones based on rearrangement theorems. The effect of convolution on correlation is also studied. Convolution makes arbitrary correlations less extreme while convolution of identical measures on R² makes extreme correlations more extreme. Extreme correlations have applications in data analysis and variance reduction in Monte Carlo studies, especially in the technique of antithetic variates.