Consider a random vector , where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and is a deterministic 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 based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for and . 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 , 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.
Bernoulli
30(1):
346-370
(February 2024).
DOI: 10.3150/23-BEJ1600
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