Logarithmic law of large random correlation matrix

03/25/2021
āˆ™
by   Nestor Parolya, et al.
āˆ™
0
āˆ™

Consider a random vector š²=Ī£^1/2š±, where the p elements of the vector š± are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Ī£^1/2 is a deterministic pƗ p matrix such that the spectral norm of the population correlation matrix š‘ of š² is uniformly bounded. In this paper, we find that the log determinant of the sample correlation matrix š‘Ģ‚ based on a sample of size n from the distribution of š² satisfies a CLT (central limit theorem) for p/nā†’Ī³āˆˆ (0, 1] and pā‰¤ n. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of š² 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 š‘=šˆ, 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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset