Logarithmic law of large random correlation matrices

• Consider a random vector y=Σ1∕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 Σ1∕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 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.

Subject headings

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

Keyword

CLT

dependent data

large-dimensional asymptotic

log determinant

random matrix theory

sample correlation matrix

Publication and Content Type

ref (subject category)

art (subject category)