Logarithmic law of large random correlation matrices

Abstract

Consider a random vector $\mathbf{y}={\mathrm{\Sigma }}^{1∕2}\mathbf{x}$, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and ${\mathrm{\Sigma }}^{1∕2}$ is a deterministic $p\times p$ matrix such that the eigenvalues of the population correlation matrix R of y are uniformly bounded away from zero and infinity. In this paper, we find that the log determinant of the sample correlation matrix $\hat{\mathbf{R}}$ based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for $p∕n\to \mathit{\gamma }\in (0,1]$ and $p\le n$. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after re-centering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. Finally, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case $\mathbf{R}=\mathbf{I}$, the test statistic becomes distribution-free and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.