Copulas with prescribed correlation matrix

Abstract

Consider the convex set Rn of semi positive definite matrices of order n with diagonal (1,…,1): If m is a distribution in Rn with second moments, denote by R(µ) Î Rn its correlation matrix. Denote by Cn the set of distributions in [0,1]nwith all margins uniform on [0,1] (called copulas). The paper proves that (Formula presented.) is a surjection from Cn on Rn if n £ 9: It also studies the Gaussian copulas µ such that R (µ)= R for a given R Î Rn.