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 = (rho(ij)) be the p x p sample correlation matrix of X-n; that is, the entry rho(ij) is the usual Pearson's correlation coefficient between the i th column of X-n and j th 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(inot equalj) \rho(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.