February 2024 Logarithmic law of large random correlation matrices
Nestor Parolya, Johannes Heiny, Dorota Kurowicka
Author Affiliations +
Bernoulli 30(1): 346-370 (February 2024). DOI: 10.3150/23-BEJ1600

Abstract

Consider a random vector y=Σ12x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ12 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 Rˆ based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for pnγ(0,1] and pn. 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.

Acknowledgments

J. Heiny’s research was supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity. The authors thank Gabriella F. Nane for fruitful discussions.

Citation

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Nestor Parolya. Johannes Heiny. Dorota Kurowicka. "Logarithmic law of large random correlation matrices." Bernoulli 30 (1) 346 - 370, February 2024. https://doi.org/10.3150/23-BEJ1600

Information

Received: 1 August 2022; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665581
zbMATH: 07788887
Digital Object Identifier: 10.3150/23-BEJ1600

Keywords: CLT , dependent data , large-dimensional asymptotic , log determinant , Random matrix theory , sample correlation matrix

Vol.30 • No. 1 • February 2024
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