## 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.

### MSC:

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