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.

In this paper, we show the central limit theorem for the logarithmic determinant of the sample correlation matrix R constructed from the (p × n)-dimensional data matrix X containing independent and identically distributed random entries with mean zero, variance one and infinite fourth moments. Precisely, we show that for p/n → γ ∈ (0, 1) as n, p → ∞ the logarithmic law log det R − (p − n + ¹ ₂ )log(1 − p/n) + p − p/n _{→d N(0, 1)} ^√−2 log(1 − p/n) − 2p/n is still valid if the entries of the data matrix X follow a symmetric distribution with a regularly varying tail of index α ∈ (3, 4). The latter assumptions seem to be crucial, which is justified by the simulations: if the entries of X have the infinite absolute third moment and/or their distribution is not symmetric, the logarithmic law is not valid anymore. The derived results highlight that the logarithmic determinant of the sample correlation matrix is a very stable and flexible statistic for heavy-tailed big data and open a novel way of analysis of high-dimensional random matrices with self-normalized entries.