Copulas in three dimensions with prescribed correlations

Abstract

Given an arbitrary three-dimensional correlation matrix, we prove that there exists a three-dimensional joint distribution for the random variable $(X,Y,Z)$ such that $X$,$Y$ and $Z$ are identically distributed with beta distribution $\beta_{k,k}(dx)$ on $(0,1)$ if $k\geq 1/2$. This implies that any correlation structure can be attained for three-dimensional copulas.