## Abstract

Consider a random vector y = Σ
^{1/2} x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ
^{1/2} is a deterministic p × p matrix such that the eigenvalues of the population correlation matrix R of y are uniformly bounded away from zero and infinity. In this paper, we find that the log determinant of the sample correlation matrix R based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for p/n → γ ∈ (0, 1] and p ≤ n. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after re-centering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. Finally, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case R = I, the test statistic becomes distribution-free and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

Original language | English |
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Pages (from-to) | 346 - 370 |

Number of pages | 24 |

Journal | Bernoulli |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2024 |

### Bibliographical note

Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

## Keywords

- CLT
- dependent data
- large-dimensional asymptotic
- log determinant
- random matrix theory
- sample correlation matrix