2021/03/25 by Nestor Parolya, Parolya, Nestor, Johannes Heiny +3 · 1 citation
Mathematics · #60B20 #60F05 #60F15 #60F17 #62H10 #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2103.13900
openalex publication_date 2021/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a random vector y=\mathbfΣ1/2x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and \mathbfΣ1/2 is a deterministic p× p matrix such that the spectral norm of the population correlation matrix R of y is uniformly bounded. 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 recentering 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. At last, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case R=I, the test statistic becomes completely pivotal 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.