Correlation Tests and Linear Spectral Statistics of the Sample Correlation Matrix

Testing the independence of the entries of multidimensional Gaussian observations is a very important problem in statistics, with a number of applications in signal processing, radar, cognitive radio, seismography, and multiple other fields. Typically, the problem is formulated as a binary hypothesis test, whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold. Most of the statistics for correlation tests are constructed from the sample correlation matrix (also known as sample coherence matrix in signal processing), which is defined as a power-normalized version of the sample covariance matrix. In this paper, correlation tests constructed from linear spectral statistics (LSS) of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude. A central limit theorem (CLT) is established on this class of statistics, which is valid for generally correlated Gaussian observations. Results show that LSS asymptotically fluctuate as Gaussian random variables under both the hypotheses, with an asymptotic mean and variance that can be established for each particular test. In particular, this general CLT can be used to establish the asymptotic behavior of two of the most important correlation test statistics, namely the generalized likelihood ratio test and the Frobenius norm test, under both null and alternative hypotheses. As a by-product, it is established that LSS of sample covariance and sample correlation matrices have exactly the same first order behavior, but quite different asymptotic fluctuations in the second-order regime. In both the cases, the LSS asymptotically behave as Gaussian random variables, although with quite different asymptotic means and variances.