Copulas with Prescribed Correlation Matrix

Abstract

Consider the convex set R _n of semi positive definite matrices of order n with diagonal $(1,\ldots,1).$ If μ is a distribution in $\mathbb{R}^{n}$ with second moments, denote by R(μ) ∈ R _n its correlation matrix. Denote by C _n the set of distributions in [0, 1]ⁿ with all margins uniform on [0, 1] (called copulas). The paper proves that $\mu \mapsto R(\mu )$ is a surjection from C _n on R _n if n ≤ 9. It also studies the Gaussian copulas μ such that R(μ) = R for a given R ∈ R _n .