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**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.

Reviewer: Kévin Allan Sales Rodrigues (São Paulo)

### 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 |

### Keywords:

maximum sample correlation; phase transition; multivariate normal distribution; Gumbel distribution; Chen-Stein Poisson approximation
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\textit{J. Fan} and \textit{T. Jiang}, Ann. Probab. 47, No. 5, 3321--3374 (2019; Zbl 1435.62077)

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