# Does a maximal coupling distribution have the highest correlation?

Sampling from a maximal coupling of 2 univariate Gaussians will give you a bivariate distribution which is not bivariate Gaussian. A bivariate Gaussian can be parameterized with a correlation parameter $\rho$. I assume that the maximal coupling distribution will have a larger correlation than the bivariate Gaussian but I might be wrong on this-the maximal coupling could be less correlated than the bivariate Gaussian. Are there any results on the relationship between maximal coupling and correlation?

• The question does not make complete sense in that there is no $\rho$ involved in the statement of the maximal coupling, only marginals matter. – Xi'an Dec 25 '17 at 10:07
• @Xi'an I agree the the marginals are explicitly specified but the procedure mentioned on Pierre's blog also gives an implied joint distribution over $(X,Y)$ does it not? I was thinking for the normal case you might be able to use a conditioning argument to get to a relationship between covariance of the implied distribution over $(X,Y)$ under the 2 cases in Pierre's algorithm. From this you could get a correlation. Incidentally, I was reading Pierre's blog post and came up with this question-I should've included a link in the original post-thanks for including. – Lucas Roberts Dec 26 '17 at 16:29

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.
Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):