## Largest entries of sample correlation matrices from equi-correlated normal populations.(English)Zbl 1435.62077

The authors proved two important theorems about the limiting distributions of the largest entry outside the diagonal of the sample covariance / correlation matrices generated by a random sample from a high-dimensional normal distribution, assuming that: the normal distribution has the equi-correlation structure (with parameter $$\rho$$), that $$p \to \infty$$ and $$\log(p) = o(n^{1/3})$$, where $$p$$ is the random vector dimension and $$n$$ is the number of observations. The theory developed is exemplified by means of an application to a statistical test involving a high-dimensional normal distribution (testing if the covariance matrix is diagonal). Finally, in the “discussions” subsection the authors conclude that behaviors of limiting distributions depend on the value of $$\rho$$. The limits are the normal distribution if $$\rho$$ is reasonably large; the limiting distributions are the extreme-value distribution if $$\rho$$ is tiny. Moreover, the authors also figure out the regime to differentiate the two scenarios.

### MSC:

 62E20 Asymptotic distribution theory in statistics 62H10 Multivariate distribution of statistics 62H15 Hypothesis testing in multivariate analysis 60F05 Central limit and other weak theorems 62G32 Statistics of extreme values; tail inference
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