Logarithmic law of large random correlation matrix
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Consider a random vector 𝐲=Σ^1/2𝐱, where the p elements of the vector 𝐱 are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ^1/2 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 𝐑̂ 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.
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