Logarithmic law of large random correlation matrices

Consider a random vector y = Σ ^1/2 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 Σ ^1/2 is a deterministic p × 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 R based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for p/n → γ ∈ (0, 1] and p ≤ 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 R = 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.