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Following my post of yesterday about the missing condition in Lynch’s R code, Gérard Letac sent me a paper he recently wrote with Luc Devroye on correlation matrices and copulas. Paper written for the memorial volume in honour of Marc Yor. It considers the neat problem of the existence of a copula (on [0,1]x…x[0,1]) associated with a given correlation matrix R. And establishes this existence up to dimension n=9. The proof is based on the consideration of the extreme points of the set of correlation matrices. The authors conjecture the existence of (10,10) correlation matrices that cannot be a correlation matrix for a copula. The paper also contains a result that answers my (idle) puzzling of many years, namely on how to set the correlation matrix of a Gaussian copula to achieve a given correlation matrix R for the copula. More precisely, the paper links the [correlation] matrix R of X~N(0,R) with the [correlation] matrix R⁰ of Φ(X) by
A side consequence of this result is that there exist correlation matrices of copulas that cannot be associated with Gaussian copulas. Like
![R=\left[\begin{matrix} 1 &-1/2 &-1/2\\-1/2 &1 &-1/2\\-1/2 &-1/2 &1 \end{matrix}\right]](https://s0.wp.com/latex.php?latex=R%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D+1+%26-1%2F2+%26-1%2F2%5C%5C-1%2F2+%261+%26-1%2F2%5C%5C-1%2F2+%26-1%2F2+%261+%5Cend%7Bmatrix%7D%5Cright%5D&bg=000000&%23038;fg=B0B0B0&%23038;s=0 "R=\left[\begin{matrix} 1 &-1/2 &-1/2\\-1/2 &1 &-1/2\\-1/2 &-1/2 &1 \end{matrix}\right]")
