Correlation test for high dimensional data with application to signal detection in sensor networks

The problem of correlation detection of multivariate Gaussian observations is considered. The problem is formulated as a binary hypothesis test, where the null hypothesis corresponds to a diagonal correlation matrix with possibly different diagonal entries, whereas the alternative would be associated to any other form of positive covariance. Using tools from random matrix theory, we study the asymptotic behavior of the Generalized Likelihood Ratio Test (GLRT) under both hypothesis, assuming that both the sample size and the observation dimension tend to infinity at the same rate. It is shown that the GLRT statistic always converges to a Gaussian distribution, although the asymptotic mean and variance will strongly depend the actual hypothesis. Numerical simulations demonstrate the superiority of the proposed asymptotic description in situations where the sample size is not much larger than the observation dimension.