The asymptotic distributions of the largest entries of sample correlation matrices

Let X _n = (x _ij) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R _n = (ρ _ij) be the p × p sample correlation matrix of X _n; that is, the entry ρ _ij is the usual Pearson's correlation coefficient between the ith column of X _n and jth column of X _n. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H ₀ : the p variates of the population are uncorrelated. A test statistic is chosen as L _n = max _{i ≠ j} |ρ _ij|. The asymptotic distribution of L _n is derived by using the Chen-Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived.