The asymptotic distributions of the largest entries of sample correlation matrices. (English) Zbl 1047.60014

Summary: 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 \(p \times p\) 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 the \(j\)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 \(L_n=\max_{i\neq j}| \rho_{ij} |\). 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.


60F05 Central limit and other weak theorems
60F15 Strong limit theorems
62H10 Multivariate distribution of statistics
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