A0941
Title: Logarithmic law for large sample correlation matrices
Authors: Nestor Parolya - Delft University of Technology (Netherlands) [presenting]
Johannes Heiny - Ruhr University Bochum (Germany)
Dorota Kurowicka - Delft University of Technology (Netherlands)
Abstract: The log determinant of the sample correlation matrix based on a data matrix of size $n$ and dimension $p$ is found to satisfy a CLT (central limit theorem) for $p/n$ in $(0,1]$ and $p\le n$. Explicit formulas for the asymptotic mean and variance are provided. In case the population mean 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 together with the new theory 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.
