### Abstract

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} = (ρ _{ij}) be the p × p sample correlation matrix of X _{n}; that is, the entry ρ _{ij} is the usual Pearson's correlation coefficient between the ith column of X _{n} and jth 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 ≠ j} |ρ _{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.

Original language | English (US) |
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Pages (from-to) | 865-880 |

Number of pages | 16 |

Journal | Annals of Applied Probability |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - May 2004 |

### Keywords

- Chen-Stein method
- Maxima
- Moderate deviations
- Sample correlation matrices